Computational Geosciences

, Volume 17, Issue 2, pp 373–390 | Cite as

Meshless numerical modeling of brittle–viscous deformation: first results on boudinage and hydrofracturing using a coupling of discrete element method (DEM) and smoothed particle hydrodynamics (SPH)

Original Paper

Abstract

We have developed a new approach for the numerical modeling of deformation processes combining brittle fracture and viscous flow. The new approach is based on the combination of two meshless particle-based methods: the discrete element method (DEM) for the brittle part of the model and smooth particle hydrodynamics (SPH) for the viscous part. Both methods are well established in their respective application domains. The two methods are coupled at the particle scale, with two different coupling mechanisms explored: one is where DEM particles act as virtual SPH particles and one where SPH particles are treated like DEM particles when interacting with other DEM particles. The suitability of the combined approach is demonstrated by applying it to two geological processes, boudinage, and hydrofracturing, which involve the coupled deformation of a brittle solid and a viscous fluid. Initial results for those applications show that the new approach has strong potential for the numerical modeling of coupled brittle–viscous deformation processes.

Keywords

Numerical modeling Coupled brittle–ductile deformation DEM SPH Boudinage  Hydrofracturing 

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References

  1. 1.
    Abe, S.: Investigation of the influence of different micro-physics on the dynamic behaviour of faults using the lattice solid model. Ph.D. thesis, The University of Queensland, Brisbane (2002)Google Scholar
  2. 2.
    Abe, S., Dieterich, J.H., Mora, P., Place, D.: Simulation of the influence of rate- and state-dependent friction on the macroscopic behavior of complex fault zones with the lattice solid model. Pure Appl. Geophys. 159(9), 1967–1983 (2002)CrossRefGoogle Scholar
  3. 3.
    Abe, S., van Gent, H., Urai, J.L.: DEM simulation of normal faults in cohesive materials. Tectonophysics 512(1–4), 12–21 (2011)CrossRefGoogle Scholar
  4. 4.
    Abe, S., Latham, S., Gross, L., Smilie, J.: Coupling finite elements and particle based simulation. In: Papadrakakis, M., Onate, E., Schrefler, B. (eds.) Int. Conf. on Computational Methods for Coupled Problems in Science and Engineering Coupled Problems. CIMNE, Barcelona (2005)Google Scholar
  5. 5.
    Abe, S., Latham, S., Mora, P.: Dynamic rupture in a 3-D particle-based simulation of a rough planar fault. Pure Appl. Geophys. 163, 1881–1892 (2006). doi:10.1007/s00024-006-0102-6 CrossRefGoogle Scholar
  6. 6.
    Abe, S., Mair, K.: Grain fracture in 3D numerical simulations of granular shear. Geophys. Res. Lett. 32(5) (2005)Google Scholar
  7. 7.
    Abe, S., Mair, K.: Effects of gouge fragment shape on fault friction: new 3D modelling results. Geophys. Res. Lett. L23302 (2009). doi:10.1029/2009GL040684
  8. 8.
    Abe, S., Place, D., Mora, P.: A parallel implementation of the lattice solid model for the simulation of rock mechanics and earthquake dynamics. Pure Appl. Geophys. 161(11–12), 2265–2277 (2004)Google Scholar
  9. 9.
    Abe, S., Urai, J.L.: Discrete element modeling of boudinage: insights on rock rheology, matrix flow, and evolution of geometry. J. Geophys. Res. Solid Earth 117, B01407 (2012)CrossRefGoogle Scholar
  10. 10.
    Adachi, A., Siebrits, E., Peirce, A., Desroches, J.: Computer simulation of hydraulic fractures. Int. J. Rock Mech. Min. Sci. 44(5), 739–757 (2007)CrossRefGoogle Scholar
  11. 11.
    Akulich, A., Zvyagin, A.: Numerical simulation of hydraulic fracture crack propagation. Moscow Univ. Mech. Bull. 63(1), 6–12 (2008)Google Scholar
  12. 12.
    Al-Busaidi, A., Hazzard, J.F., Young, R.P.: Distinct element modeling of hydraulically fractured lac du bonnet granite. J. Geophys. Res. Solid Earth 110(B6) (2005)Google Scholar
  13. 13.
    Allen, M.P., Tildesley, D.J.: Computer Simulation of Liquids. Oxford Science Publications, Oxford (1987)Google Scholar
  14. 14.
    Behnia, M., Goshtasbi, K., Golshani, A., Marji, M.F.: Numerical modeling of artificial hydro-fractures in hot dry rock reservoirs by using displacement discontinuty method. In: Proceedings, Thirty-Sixth Workshop on Geothermal Reservoir Engineering, pp. 344–355. Stanford University, Stanford, California, 31 January–2 February 2011, SGP-TR-191, ISBN: 978-1-61782-788-4 (2011)Google Scholar
  15. 15.
    Bruno, M.S., Dorfmann, L., Lao, K., Honeger, C.: Coupled particle and fluid flow modeling of fracture and slurry injection in weakly consolidated granular media. In: International Conference on Trends in Computational Structural Mechanics, pp. 647–659. Schloss Hofen, Austria, 20–23 May 2001 (2001)Google Scholar
  16. 16.
    Carter, B., Desroches, J., Ingraffea, A.R., Wawrzynek, P.A.: Simulating Fully 3D Hydraulic Fracturing. Wiley, New York (2000)Google Scholar
  17. 17.
    Clark, J.B.: A hydraulic process for increasing the productivity of wells. Trans. Am. Inst. Min. Metall. Eng. 186(1), 1–8 (1949)Google Scholar
  18. 18.
    Cundall, P.A., Strack, O.D.L.: Discrete numerical-model for granular assemblies. Geotechnique 29(1), 47–65 (1979)CrossRefGoogle Scholar
  19. 19.
    Donze, F., Mora, P., Magnier, S.A.: Numerical simulation of faults and shear zones. Geophys. J Int. 116(1), 46–52 (1994). doi:10.1111/j.1365-246X.1994.tb02126.x CrossRefGoogle Scholar
  20. 20.
    Flekkoy, E., Malte-Sorenssen, A., Jamsveit, B.: Modeling hydrofracture. J. Geophys. Res 107(B8), 2151 (2002). doi:10.1029/2000JB000132 CrossRefGoogle Scholar
  21. 21.
    Gay, N.C.: Pure shear and simple shear deformation of inhomogeneous viscous fluids .2. The determination of total finite strain in a rock from objects such as deformed pebbles. Tectonophysics 5(4), 295 (1968)CrossRefGoogle Scholar
  22. 22.
    Gay, N.C., Jaeger, J.C.: Cataclastic deformation of geological materials in matrices of differing composition: I. pebbles and conglomerates. Tectonophysics 27(4), 303–322 (1975)CrossRefGoogle Scholar
  23. 23.
    Gemmer, L., Beaumont, C., Ings, S.J.: Dynamic modelling of passive margin salt tectonics: effects of water loading, sediment properties and sedimentation patterns. Basin Res. 17(3), 383–402 (2005)CrossRefGoogle Scholar
  24. 24.
    Gemmer, L., Ings, S.J., Medvedev, S., Beaumont, C.: Salt tectonics driven by differential sediment loading: stability analysis and finite-element experiments. Basin Res. 16(2), 199–218 (2004)CrossRefGoogle Scholar
  25. 25.
    van Gent, H., Urai, J.L., De Keijzer, M.: The internal geometry of salt structures—a first look using 3D seismic data from the zechstein of the Netherlands. Special issue: flow of rocks: field analysis and modeling - in celebration of Paul F. Williams’ contribution to mentoring J. Struct. Geol. 33, 292–311 (2011)Google Scholar
  26. 26.
    Gingold, R.A., Monaghan, J.J.: Smoothed particle hydrodynamics—theory and application to non-spherical stars. Mon. Not. R. Astron. Soc. 181(2), 375–389 (1977)Google Scholar
  27. 27.
    Goscombe, B.D., Passchier, C.W., Hand, M.P.: Boudinage classification: end-member boudin types and modified boudin structures. J. Struct. Geol. 26(4), 739–763 (2004)CrossRefGoogle Scholar
  28. 28.
    Harris, L., Koyi, H.: Centrifuge modelling of folding in high-grade rocks during rifting. J. Geophys. Res. 25(2), 291–305 (2003). doi:10.1016/S0191-8141(02)00018-4 Google Scholar
  29. 29.
    Holmes, D.W., Williams, J.R., Tilke, P.: Smooth particle hydrodynamics simulations of low reynolds number flows through porous media. Int. J. Numer. Anal. Methods Geomech. 35(4), 419–437 (2011)CrossRefGoogle Scholar
  30. 30.
    Hubbert, M., Willis, D.: Mechanics of hydraulic fracturing. Mem. Am. Assoc. Pet. Geol. (USA) 18, 239–257 (1972)Google Scholar
  31. 31.
    Ings, S.J., Beaumont, C.: Shortening viscous pressure ridges, a solution to the enigma of initiating salt ‘withdrawal’ minibasins. Geology 38(4), 339–342 (2010)CrossRefGoogle Scholar
  32. 32.
    Ishida, T.: Acoustic emission monitoring of hydraulic fracturing in laboratory and field. Constr. Build. Mater. 15, 283–295 (2001)CrossRefGoogle Scholar
  33. 33.
    Ishida, T., Chen, Q., Mizuta, Y., Roegiers, J.C.: Influence of fluid viscosity on the hydraulic fracturing mechanism. J. Energy Resour. Technol. 126(3), 190–200 (2004). doi:10.1115/1.1791651. http://link.aip.org/link/?JRG/126/190/1 CrossRefGoogle Scholar
  34. 34.
    Ismail-Zadeh, A., Tackley, P.: Computational Methods for Geodynamics. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  35. 35.
    Iyer, K., Podladchikov, Y.Y.: Transformation-induced jointing as a gauge for interfacial slip and rock strength. Earth Planet. Sci. Lett. 280, 159–166 (2009)CrossRefGoogle Scholar
  36. 36.
    Jonathan, B.: Chapter 2 Reservoir Completion, vol. 56, pp. 15–128. Elsevier (2009)Google Scholar
  37. 37.
    Kaus, B.J.P., Podladchikov, Y.Y.: Forward and reverse modeling of the three-dimensional viscous Rayleigh-Taylor instability. Geophys. Res. Lett. 28(6), 1095–1098 (2001)CrossRefGoogle Scholar
  38. 38.
    Kenis, I., Urai, J.L., van der Zee, W., Hilgers, C., Sintubin, M.: Rheology of fine-grained siliciclastic rocks in the middle crust—evidence from a combined structural and numerical analysis. Earth Planet. Sci. Lett. 233, 351–360 (2005)CrossRefGoogle Scholar
  39. 39.
    Kenis, I., Urai, J.L., van der Zee, W., Sintubin, M.: Mullions in the high-Ardenne Slate Belt (Belgium): numerical model and parameter sensitivity analysis. J. Struct. Geol. 26, 1677–1692 (2004)CrossRefGoogle Scholar
  40. 40.
    Koyi, H.A.: Modeling the influence of sinking anhydrite blocks on salt diapirs targeted for hazardous waste disposal. Geology 29(5), 387–390 (2001)CrossRefGoogle Scholar
  41. 41.
    Li, S., Abe, S., Reuning, L., Becker, S., Urai, J.L., Kukla, P.A.: Numerical modelling of the displacement and deformation of embedded rock bodies during salt tectonics: a case study from the south oman salt basin. Geol. Soc. London Spec. Publ. 363(1), 503–520 (2012)CrossRefGoogle Scholar
  42. 42.
    Liu, G.R., Liu, M.B.: Smoothed Particle Hydrodynamics: A Meshfree Particle Method. World Scientific, New Jersey (2003)CrossRefGoogle Scholar
  43. 43.
    Liu, M.B., Liu, G.R.: Smoothed particle hydrodynamics (sph): an overview and recent developments. Arch. Comput. Methods Eng. 17(1), 25–76 (2010)CrossRefGoogle Scholar
  44. 44.
    Liu, M.B., Liu, G.R., Zong, Z., Lam, K.Y.: Computer simulation of high explosive explosion using smoothed particle hydrodynamics methodology. Comput. Fluids 32(3), 305–322 (2003)CrossRefGoogle Scholar
  45. 45.
    Lucy, L.B.: Numerical approach to testing of fission hypothesis. Astron. J. 82(12), 1013–1024 (1977)CrossRefGoogle Scholar
  46. 46.
    Maeder, X., Passchier, C.W., Koehn, D.: Modelling of segment structures: boudins, bone-boudins, mullions and related single- and multiphase deformation features. J. Struct. Geol. 31(8), 817–830 (2009)CrossRefGoogle Scholar
  47. 47.
    Mair, K., Abe, S.: 3d numerical simulations of fault gouge evolution during shear: grain size reduction and strain localization. Earth Planet. Sci. Lett. 274, 72–81 (2008)CrossRefGoogle Scholar
  48. 48.
    Mair, K., Abe, S.: Breaking up: comminution mechanisms in sheared simulated fault gouge. Pure Appl. Geophys. (2011). doi:10.1007/s00024-011-0266-6
  49. 49.
    Mandal, N., Khan, D.: Rotation, offset and separation of oblique-fracture (rhombic) boudins: theory and experiments under layer-normal compression. J. Struct. Geol. 13(3), 349–356 (1991)CrossRefGoogle Scholar
  50. 50.
    McDougall, S., Hungr, O.: A model for the analysis of rapid landslide motion across three-dimensional terrain. Can. Geotech. J. 41(6), 1084–1097 (2004)CrossRefGoogle Scholar
  51. 51.
    Melean, Y., Sigalotti, L.D.G., Hasmy, A.: On the SPH tensile instability in forming liquid viscous drops. Comput. Phys. Commun. 157, 191–200 (2004). doi:10.1016/j.comphy.2003.11.002 CrossRefGoogle Scholar
  52. 52.
    Melosh, H.J., Williams, C.A.: Mechanics of graben formation in crustal rocks: a finite element analysis. J. Geophys. Res. 94(B10), 13,961–13,973 (1989). doi:10.1029/JB094iB10p13961 CrossRefGoogle Scholar
  53. 53.
    Monaghan, J.J.: Smoothed particle hydrodynamics. Annu. Rev. Astron. Astrophys. 30, 543–574 (1992)CrossRefGoogle Scholar
  54. 54.
    Monaghan, J.J.: Simulating free surface flows with sph. J. Comput. Phys. 110(2), 399–406 (1994)CrossRefGoogle Scholar
  55. 55.
    Monaghan, J.J.: SPH without a tensile instablity. J. Comp. Phys. 159, 290–311 (2000). doi:10.1006/jcph.2000.6439 CrossRefGoogle Scholar
  56. 56.
    Monaghan, J.J., Kocharyan, A.: SPH simulation of multiphase flow. Comput. Phys. Commun. 87(1–2), 225–235 (1995)CrossRefGoogle Scholar
  57. 57.
    Mora, P., Place, D.: Simulation of the frictional stick-slip instability. Pure Appl. Geophys. 143(1–3), 61–87 (1994)CrossRefGoogle Scholar
  58. 58.
    Mora, P., Place, D.: Numerical simulation of earthquake faults with gouge: towards a comprehensive explanation for the heat flow paradox. J. Geophys. Res. 103, 21,067–21,089 (1998)CrossRefGoogle Scholar
  59. 59.
    Morgan, J.K.: Numerical simulations of granular shear zones using the distinct element method, 1. Shear zone kinematics and the micromechanics of localization. J. Geophys. Res. 104(B2), 2703–2718 (1999)CrossRefGoogle Scholar
  60. 60.
    Morris, J.P., Fox, P.J., Zhu, Y.: Modeling low reynolds number incompressible flows using SPH. J. Comput. Phys. 136(1), 214–226 (1997)CrossRefGoogle Scholar
  61. 61.
    Neurath, C., Smith, R.: The effect of material properties on growth rates of folding and boudinage: experiments with wax models. J. Struct. Geol. 4(2), 215–229 (1982). doi:10.1016/0191-8141(82)90028-1 CrossRefGoogle Scholar
  62. 62.
    Nieuwland, D., Urai, J., Knoop, M.: In-situ stress measurements in model experiments of tectonic faulting. In: F. Lehner, Urai, J. (eds.) Aspects of Tectonic Faulting: In Honour of Georg Mandl, pp. 151–162. Springer, Berlin (2000)Google Scholar
  63. 63.
    Nordgren, R.P.: Propagation of a vertical hydraulic fracture. J. Pet. Technol. 22,1052 (1970)Google Scholar
  64. 64.
    Pak, A., Chan, D.H.: Numerical modeling of hydraulic fracturing in oil sands. Sci. Iran. 15(5), 20 (2008)Google Scholar
  65. 65.
    Passchier, C.W., Druguet, E.: Numerical modelling of asymmetric boudinage. J. Struct. Geol. 24(11), 1789–1803 (2002)CrossRefGoogle Scholar
  66. 66.
    Perkins, T.K., Kern, L.R.: Widths of hydraulic fractures. Trans. Soc. Pet. Eng. Am. 222(9), 937–949 (1961)Google Scholar
  67. 67.
    Place, D., Mora, P.: The lattice solid model: incorporation of intrinsic friction. J. Int. Comp. Phys. 150, 332–372 (1999)CrossRefGoogle Scholar
  68. 68.
    Podladchikov, Y., Talbot, C., Poliakov, A.N.B.: Numerical-models of complex diapirs. Tectonophysics 228(3–4), 189–198 (1993)CrossRefGoogle Scholar
  69. 69.
    Poliakov, A.N.B., Vanbalen, R., Podladchikov, Y., Daudre, B., Cloetingh, S., Talbot, C.: Numerical-analysis of how sedimentation and redistribution of surficial sediments affects salt diapirism. Tectonophysics 226(1–4), 199–216 (1993)Google Scholar
  70. 70.
    Potyondy, D.O., Cundall, P.A.: A bonded-particle model for rock. Int. J. Rock Mech. Min. Sci. 41(8), 1329–1364 (2004)CrossRefGoogle Scholar
  71. 71.
    Ramberg, H.: Natural and experimental boudinage and pinch-and-swell structures. J. Geol. 63(6), 512 (1955)CrossRefGoogle Scholar
  72. 72.
    Rosswog, S., Wagner, P.: Towards a macroscopic modeling of the complexity in traffic flow. Phys. Rev. E 65(3) (2002)Google Scholar
  73. 73.
    Rutledge, J.T., Phillips, W.S., House, L.S., Zinno, R.J.: Microseismic mapping of a cotton valley hydraulic fracture using decimated downhole arrays. In: Expanded Abstracts SEG Annual Meeting, pp. 338–341 (1998)Google Scholar
  74. 74.
    Sasaki, S.: Characteristics of microseismic events induced during hydraulic fracturing experiments at the Hijiori hot dry rock geothermal energy site, Yamagata, Japan. Tectonophysics 289(1–3), 171–188 (1998)CrossRefGoogle Scholar
  75. 75.
    Schenk, O., Urai, J.L., van der Zee, W.: Evolution of boudins under progressively decreasing pore pressure—a case study of pegmatites enclosed in marble deforming at high grade metamorphic conditions, Naxos, Greece. Am. J. Sci. 307(7), 1009–1033 (2007)CrossRefGoogle Scholar
  76. 76.
    Schmatz, J., Vrolijk, P., Urai, J.L.: Clay smear in normal fault zones: the effect of multilayers and clay cementation in water-saturated model experiments. J. Struct. Geol. 32(11), 1834–1849 (2010)CrossRefGoogle Scholar
  77. 77.
    Schoenherr, J., Reuning, L., Kukla, P.A., Littke, R., Urai, J.L., Siemann, M., Rawahi, Z.: Halite cementation and carbonate diagenesis of intra-salt reservoirs from the late neoproterozoic to Early Cambrian Ara Group (South Oman Salt Basin). Sedimentology 56(2), 567–589 (2009)CrossRefGoogle Scholar
  78. 78.
    Schoepfer, M.P.J., Abe, S., Childs, C., Walsh, J.J.: The impact of porosity and crack density on the elasticity, strength and friction of cohesive granular materials: insights from DEM modelling. Int. J. Rock Mech. Min. Sci. 46(2), 250–261 (2009)CrossRefGoogle Scholar
  79. 79.
    Schoepfer, M.P.J., Arslan, A., Walsh, J.J., Childs, C.: Reconciliation of contrasting theories for fracture spacing in layered rocks. J. Geophys. Res. 33(551–565) (2011). doi:10.1016/j.jsg.2011.01.008
  80. 80.
    Schultz-Ela, D.D., Walsh, P.: Modeling of grabens extending above evaporites in Canyonlands National Park, Utah. J. Struct. Geol. 24(2), 247–275 (2002)CrossRefGoogle Scholar
  81. 81.
    Shimizu, H., Murata, S., Ishida, T.: The distinct element analysis for hydraulic fracturing in hard rock considering fluid viscosity and particle size distribution. Int. J. Rock Mech. Min. Sci. 48(5), 712–727 (2011)CrossRefGoogle Scholar
  82. 82.
    Strozyk, F., van Gent, H.W., Urai, J.L., Kukla, P.A.: 3D seismic study of complex intra-salt deformation: an example from the upper Permian Zechstein 3 stringer, western Dutch offshore. In: Alsop, G.I., Archer, S.G., Hartley, A.J., Grant, N.T., Hodgkinson, R., (eds.) Salt Tectonics, Sediments and Prospectivity, vol. 363, pp. 489–501. Geological Society of London, Special Publication (2012)Google Scholar
  83. 83.
    Swegle, J.W., Hicks, D.L., Attaway, S.W.: Smoothed particle hydrodynamics stability analysis. J. Comp. Phys. 116, 123–134 (1995)CrossRefGoogle Scholar
  84. 84.
    Takeda, H., Miyama, S.M., Sekiya, M.: Numerical simulation of viscous flow by smoothed particle hydrodynamics. Prog. Theor. Phys. 92(5), 939–960 (1994)CrossRefGoogle Scholar
  85. 85.
    Tanaka, N., Takano, T.: Microscopic-scale simulation of blood flow using SPH method. Int. J. Comput. Methods 2(4), 555–568 (2005)CrossRefGoogle Scholar
  86. 86.
    Thornton, C.: Numerical simulations of deviatoric shear deformation of granular media. Géotechnique 50(1), 43–53 (2000)CrossRefGoogle Scholar
  87. 87.
    Treagus, S.H., Lan, L.: Deformation of square objects and boudins. J. Struct. Geol. 26(8), 1361–1376 (2004)CrossRefGoogle Scholar
  88. 88.
    Twiss, R.J., Moores, E.M.: Structural Geology, 2nd edn. W.H. Freeman, New York (2007)Google Scholar
  89. 89.
    Urai, J.L., Schleder, Z., Spiers, C.J., Kukla, P.A.: Flow and transport properties of salt rocks. In: Littke, R., Bayer, U., Gajewski, D., Nelskamp, S. (eds.) Dynamics of Complex Intracontinental Basins: The Central European Basin System, pp. 277–290. Springer, Berlin (2008)Google Scholar
  90. 90.
    Victor, P., Moretti, I.: Polygonal fault systems and channel boudinage: 3D analysis of multidirectional extension in analogue sandbox experiments. Mar. Pet. Geol. 23, 777–789 (2006)CrossRefGoogle Scholar
  91. 91.
    Wang, S.Y., Sun, L., Au, A.S.K., Yang, T.H., Tang, C.A.: 2D-numerical analysis of hydraulic fracturing in heterogeneous geo-materials. Constr. Build. Mater. 23(6), 2196–2206 (2009)CrossRefGoogle Scholar
  92. 92.
    Wang, Y., Abe, S., Latham, S., Mora, P.: Implementation of particle-scale rotation in the 3-D lattice solid model. Pure Appl. Geophys. 163, 1769–1785 (2006)CrossRefGoogle Scholar
  93. 93.
    Weijermars, R.: Polydimethylsiloxane flow defined for experiments in fluid dynamics. Appl. Phys. Lett. 48(2), 109–111 (1986)CrossRefGoogle Scholar
  94. 94.
    Woidt, W.D., Neugebauer, H.J.: Finite-element models of density instabilities by means of bicubic spline interpolation. Phys. Earth Planet. Inter. 21, 176–180 (1980)CrossRefGoogle Scholar
  95. 95.
    Zulauf, G., Zulauf, J., Bornemann, O., Brenker, F.E., Hofer, H.E., Peinl, M., Woodland, A.B.: Experimental deformation of a single-layer anhydrite in halite matrix under bulk constriction. part 2: deformation mechanisms and the role of fluids. J. Struct. Geol. 32(3), 264–277 (2010)CrossRefGoogle Scholar
  96. 96.
    Zulauf, J., Zulauf, G.: Coeval folding and boudinage in four dimensions. J. Struct. Geol. 27(6), 1061–1068 (2005)CrossRefGoogle Scholar
  97. 97.
    Zuorong, C.: Finite element modelling of viscosity-dominated hydraulic fractures. J. Pet. Sci. Eng. 88–89, 136–144 (2012)Google Scholar

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© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Andrea Komoróczi
    • 1
  • Steffen Abe
    • 1
  • Janos L. Urai
    • 1
  1. 1.Structural Geology - Tectonics - GeomechanicsRWTH Aachen UniversityAachenGermany

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