Analysis and Numerical Modeling of Faults Using the Boundary Element Method

Document Type : Research Article

Authors

1 Ph.D. Student in Geotechnical Engineering, Department of Civil Engineering, Razi University, Kermanshah, Iran

2 Assistant Professor, Department of Civil Engineering, Razi University, Kermanshah, Iran

3 Professor, Geotechnical Engineering Research Center, International Institute of Earthquake Engineering and Seismology (IIEES), Tehran, Iran

Abstract

One of the numerical methods in the mechanics of continuous environments is the boundary element method. In this method, the governing differential equations will be converted to integral equations and applied to the problem boundary. Then the boundary is divided into boundary parts and numerical integration is performed on the boundary elements, from the solution of which a single solution of the problem can be obtained. In the boundary element method, the partial differential equations defined within a space are converted to integral equations at the boundaries of that space, which reduces one dimension of the problem. For example, an elastodynamic problem defined in a two-dimensional space is replaced by an integral equation problem defined at its boundaries that has one dimension. If the problem defined in a three-dimensional space is replaced by a two-dimensional integral equation problem. Finally, the integral equations will be solved numerically by dividing the boundaries into a network of finite element discrete elements. The boundary element method can be easily applied to borders with complex geometry. Boundary element method or boundary integral equation (BIEM) is one of the numerical modeling methods that have many applications in numerical simulation of fault dynamics. Its results provide a broad view of the physics of earthquake rupture. To solve two-dimensional problems, the numerical technique of the boundary element method has been widely used. The boundary element method has been used to model the behavior of faults of overlapping centers, the growth of junction assemblies, veins and seismic analysis of topographic features. One form of BEM is based on separation, which is called the displacement discontinuity method. The theory of detachments in elastic materials has been widely used for more than half a century to evaluate the displacement, stress and strain fields around faults. By integrating Green's functions, the displacement field around the discontinuity surface can be calculated. These displacement fields are the Navier equations that are the governing equations of linear elastic theory. Strain components are obtained from the spatial derivatives of the displacement components, and the stress components can be calculated using Hooke's law for homogeneous and homogeneous elastic materials. Therefore, the mathematical tool of detachment theory is able to calculate the displacement, stress and strain fields around faults in half-elastic space, but it is less accurate compared to geophysical data. In this paper, numerical modeling of faults using the boundary element method has been reviewed, and studies conducted in the field of numerical modeling of faults using the boundary element method have been reviewed. Finally, the results show that the boundary element method is suitable for problems with complex boundaries such as fault geometry and problems with infinite boundaries. It is also possible to predict slip on the fault and surface deformation using numerical modeling using the boundary element method. The results of numerical modeling of the fault using the boundary element method and comparing these results with the observed values ​​show that the boundary element method is a suitable method for predicting issues such as slip distribution on the fault, surface displacement and slope instability. Also, the boundary element method is a suitable method for predicting the location of smaller and secondary faults. Therefore, it can be said that the boundary element method is a powerful tool for numerical modeling of earthquake or fault rupture dynamics. In the articles studied in this study, the slip distribution at the fault surface has been determined using the inversion of surface displacements resulting from the main earthquake. If aftershocks also cause surface displacements, then as a suggestion for future research, the inversion of surface displacements caused by aftershocks can also be used to determine the secondary slips created on the fault surface.

Keywords

References

  1. Beskos, D.E. (1987) Boundary element methods in dynamic analysis. Mech. Rev., 40, 1-23.
  2. Beskos, D.E. (1997) Boundary element methods in dynamic analysis: Part II (1986-1996). Mech. Rev., 50, 149-197.
  3. Bouchon, M. and Sa´nchez-Sesma, F.J. (2007) Boundary integral equations and boundary elements methods in elastodynamics. Geophys., 48, 157-189.
  4. Zhao, S. and Chao, D. (1995) Determination of the fault slip distribution of the 1976 Tangshan earthquake by the finite element method. Journal of Geodynamics, 19(2), 107-115.‏
  5. Trasatti, E., Kyriakopoulos, C., and Chini, M. (2011) Finite element inversion of DInSAR data from the Mw 6.3 L'Aquila earthquake, 2009 (Italy). Geophysical Research Letters, 38(8).‏
  6. Zhang, W., Iwata, T. & Irikura, K. (2006) Dynamic simulation of a dipping fault using a three‐dimensional finite difference method with nonuniform grid spacing. Journal of Geophysical Research: Solid Earth, 111(B5).‏
  7. Coates, R.T. and Schoenberg, M. (1995) Finite-difference modeling of faults and fractures. Geophysics, 60(5), 1514-1526.
  8. Okada, Y. (1985) Surface deformation due to shear and tensile faults in a half-space. Bulletin of the Seismological Society of America, 75(4), 1135-1154.‏
  9. Okada, Y. (1992) Internal deformation due to shear and tensile faults in a half-space. Bulletin of the Seismological Society of America, 82(2), 1018-1040.
  10. Thomas, A.L. (1993) Poly 3D: A Three-Dimensional, Polygonal Element, Displacement Discontinuity Boundary Element Computer Program with Applications to Fractures, Faults, and Cavities in the Earth's crust. Master's Thesis, to the Department of Geology, Stanford University.‏
  11. Maerten, F., Maerten, L., and Pollard, D.D. (2014) iBem3D, a three-dimensional iterative boundary element method using angular dislocations for modeling geologic structures. Computers and Geosciences, 72, 1-17.
  1. Caniven, Y., Dominguez, S., Soliva, R., Peyret, M., Cattin, R., and Maerten, F. (2017) Relationships between along-fault heterogeneous normal stress and fault slip patterns during the seismic cycle: Insights from a strike-slip fault laboratory model. Earth and Planetary Science Letters, 480, 147-157.‏
  2. Crouch, S.L. and Starfield, A.M. (1983) Boundary Element Methods in Solid Mechanics. Unwin Hyman,
  3. Mavko, G.M. (1982) Fault interaction near Hollister, California. Journal of Geophysical Research: Solid Earth, 87(B9), 7807-7816.
  4. Bilham, R. and King, G. (1989) The morphology of strike‐slip faults: Examples from the San Andreas Fault, California. Journal of Geophysical Research: Solid Earth, 94(B8), 10204-10216.
  5. Sempere, J.C. and Macdonald, K.C. (1986) Over-lapping spreading centers: Implications from crack growth simulation by the displacement discon-tinuity method. Tectonics, 5(1), 151-163.
  6. Olson, J. and Pollard, D.D. (1989) Inferring paleostresses from natural fracture patterns: A new method. Geology, 17(4), 345-348.‏
  7. Wu, H. and Pollard, D.D. (1995) An experimental study of the relationship between joint spacing and layer thickness. Journal of Structural Geology, 17(6), 887-905.‏
  8. Olson, J.E. and Pollard, D.D. (1991) The initiation and growth of en echelon veins. Journal of Structural Geology, 13(5), 595-608.
  9. Panji, M., Kamalian, M., Asgari Marnani, J., and Jafari, M.K. (2013) Transient analysis of wave propagations problems by half-plane BEM. Geophysical Journal International, 194, 1849-1865.
  10. Panji, M., Kamalian, M., Asgari Marnani, J., and Jafari, M.K. (2014a) Analyzing seismic convex topographies by a half-plane time-domain BEM. Geophysical Journal International, 197(1), 591-607.
  11. Panji, M., Kamalian, M., Asgari Marnani, J., and Jafari, M.K. (2014b) Antiplane seismic response from semi-sine shaped valley above embedded truncated circular cavity: a half-plane time-domain BEM. IJCE, 12(2), 193-206.
  12. Steketee, J.A. (1958) On Volterra's dislocations in a semi-infinite elastic medium. Canadian Journal of Physics.36(2), 192-205.‏
  13. Steketee, J.A. (1958) Some geophysical applications of the elasticity theory of dislocations. Canadian Journal of Physics, 36(9), 1168-1198.
  14. Chinnery, M.A. (1963) The stress changes that accompany strike-slip faulting. Bulletin of the Seismological Society of America, 53(5), 921-932.‏
  15. Chinnery, M.A. (1961) The deformation of the ground around surface faults. Bulletin of the Seismological Society of America, 51(3), 355-372.‏
  16. Maruyama, T. (1964) Statical elastic dislocations in an infinite and semi-infinite medium.  Earthq. Res. Inst., 42, 289-368.‏
  17. Press, F. (1965) Displacements, strains, and tilts at teleseismic distances. Journal of Geophysical Research, 70(10), 2395-2412.‏
  18. Savage, J.C. and Hastie, L.M. (1966) Surface defor-mation associated with dip‐slip faulting. Journal of Geophysical Research, 71(20), 4897-4904.‏
  19. Savage, J.C. and Hastie, L.M. (1969) A dislocation model for the Fairview Peak, Nevada, earthquake. Bulletin of the Seismological Society of America, 59(5), 1937-1948.‏
  20. Smylie, L. and Mansinha, D.E. (1971) The Dis-placement Fields of Inclined Faults. Bulletin of the Seismological society of America, 61, 1433-1440.‏
  21. Davis, P.M. (1983) Surface deformation associated with a dipping hydrofracture. Journal of Geophysical Research: Solid Earth, 88(B7), 5826-5834.‏
  22. Ma, X.Q. and Kusznir, N.J. (1993) Modelling of near-field subsurface displacements for generalized faults and fault arrays. Journal of Structural Geology, 15(12), 1471-1484.‏
  23. Burgers, J. (1939) Internal strains in solids.  Acad. Science, 42, 293.‏
  24. Yoffe, E.H. (1960) The angular dislocation. Philo-sophical Magazine,5(50), 161-175.‏
  25. Eshelby, J.D., Bilby, B.A., Miller, K.J., and Willis, R. (Eds.) (1985) Fundamentals of Deformation and Fracture: Eshelby Memorial Symposium Sheffield 2-5 April 1984. Cambridge University Press.
  26. Barnett, D.M. (1985) The displacement field of a triangular dislocation loop. Philosophical Magazine,51(3), 383-387.
  27. Comninou, M. and Dundurs, J. (1975) The angular dislocation in a half space. Journal of Elasticity,  5(3-4), 203-216.‏
  28. Aki, K. and Richards, P.G. (2002) Quantitative Seismology. 2nd Edition, University Science Books, Sausalito.
  29. Ida, Y. (1972) Cohesive force across the tip of a longitudinal-shear crack and Griffith’s specific surface energy. Geophys. Res., 77, 3796-3805.
  30. Jeyakumaran, M., Rudnicki, J.W., and Keer, L.M. (1992) Modeling slip zones with triangular dislocation elements. Bulletin of the Seismological Society of America, 82(5), 2153-2169.‏‏
  31. Price, E.J. and Bürgmann, R. (2002) Interactions between the Landers and Hector Mine, California, earthquakes from space geodesy, boundary element modeling, and time-dependent friction. Bulletin of the Seismological Society of America, 92(4), 1450-1469.‏
  32. Muller, J.R., Aydin, A., and Maerten, F. (2003) Investigating the transition between the 1967 Mudurnu Valley and 1999 Izmit earthquakes along the North Anatolian fault with static stress changes. Geophysical Journal International, 154(2), 471-482.‏
  33. Muller, J.R. and Aydin, A. (2004) Rupture progression along discontinuous oblique fault sets: implications for the Karadere rupture segment of the 1999 Izmit earthquake, and future rupture in the Sea of Marmara. Tectonophysics, 391(1-4), 283-302.‏
  34. Muller, J.R., Aydin, A., and Wright, T.J. (2006) Using an elastic dislocation model to investigate static Coulomb stress change scenarios for earthquake ruptures in the eastern Marmara Sea region, Turkey. Geological Society, London, Special Publications, 253(1), 397-414.‏
  35. Fielding, E.J., Wright, T.J., Muller, J., Parsons, B.E., and Walker, R. (2004) Aseismic deformation of a fold-and-thrust belt imaged by synthetic aperture radar interferometry near Shahdad, southeast Iran. Geology, 32(7), 577-580.‏
  36. Lovely, P.J., Pollard, D.D., and Mutlu, O. (2009) Regions of reduced static stress drop near fault tips for large strike-slip earthquakes. Bulletin of the Seismological Society of America, 99(3), 1691-1704.‏
  37. Fielding, E.J., Lundgren, P.R., Bürgmann, R., and Funning, G.J. (2009) Shallow fault-zone dilatancy recovery after the 2003 Bam earthquake in Iran. Nature, 458(7234), 64-68.‏
  38. Rigo, A., De Chabalier, J.B., Meyer, B., and Armijo, R. (2004) The 1995 Kozani-Grevena (northern Greece) earthquake revisited: an im-proved faulting model from synthetic aperture radar Geophysical Journal International, 157(2), 727-736.‏
  39. Muller, J.R. and Aydin, A. (2005) Using mechanical modeling to constrain fault geometries proposed for the northern Marmara Sea. Journal of Geophysical Research: Solid Earth, 110(B3).‏
  40. Akoglu, A.M., Cakir, Z., Meghraoui, M., Belabbes, S., El Alami, S.O., Ergintav, S., and Akyüz, H.S. (2006) The 1994–2004 Al Hoceima (Morocco) earthquake sequence: Conjugate fault ruptures deduced from InSAR. Earth and Planetary Science Letters, 252(3-4), 467-480.‏
  41. Marshall, S.T., Cooke, M.L., Owen, S.E. (2008) Effects of nonplanar fault topology and mechanical interaction on fault-slip distributions in the Ventura Basin, California. Bulletin of the Seismological Society of America, 98(3), 1113-1127.‏
  42. Dair, L. and Cooke, M.L. (2009) San Andreas fault geometry through the San Gorgonio Pass, California. Geology, 37(2), 119-122.‏
  43. Bürgmann, R., Schmidt, D., Nadeau, R.M., d'Alessio, M., Fielding, E., Manaker, D., and Murray, M.H. (2000) Earthquake potential along the northern Hayward fault, California. Science, 289(5482), 1178-1182.‏
  44. Schmidt, D.A., Bürgmann, R., Nadeau, R.M., and d'Alessio, M. (2005) Distribution of aseismic slip rate on the Hayward fault inferred from seismic and geodetic data. Journal of Geophysical Research: Solid Earth, 110(B8).‏
  45. Kreemer, C., Blewitt, G., and Maerten, F. (2006) Co‐and postseismic deformation of the 28 March 2005 Nias Mw 8.7 earthquake from continuous GPS data. Geophysical Research Letters, 33(7).
  46. Zhang, L., Wu, J.C., Ge, L.L., Ding, X.L., and Chen, Y.L. (2008) Determining fault slip distri-bution of the Chi-Chi Taiwan earthquake with GPS and InSAR data using triangular dislocation Journal of Geodynamics, 45(4-5), 163-168.‏
  47. Cheng, L.W., Lee, J.C., Hu, J.C., and Chen, H.Y. (2009) Coseismic and postseismic slip distribution of the 2003 Mw = 6.5 Chengkung earthquake in eastern Taiwan: Elastic modeling from inversion of GPS data. Tectonophysics, 466(3-4), 335-343.‏
  48. Willemse, E.J. and Pollard, D.D. (2000) 'Normal fault growth: evolution of tipline shapes and slip distribution'. In: Aspects of Tectonic Faulting, Springer, Berlin, Heidelberg, 193-226.
  49. Willemse, E.J., Pollard, D.D., and Aydin, A. (1996) Three-dimensional analyses of slip distributions on normal fault arrays with consequences for fault scaling. Journal of Structural Geology, 18(2-3), 295-309.‏‏
  50. Willemse, E.J. (1997) Segmented normal faults: Correspondence between three‐dimensional mecha-nical models and field data. Journal of Geophysical Research: Solid Earth, 102(B1), 675-692.‏
  51. Crider, J.G. (2001) Oblique slip and the geometry of normal-fault linkage: mechanics and a case study from the Basin and Range in Oregon. Journal of Structural Geology, 23(12), 1997-2009.‏
  52. Maerten, L., Willemse, E.J., Pollard, D.D., and Rawnsley, K. (1999) Slip distributions on inter-secting normal faults. Journal of Structural Geology, 21(3), 259-272.‏
  53. Maerten, L. (1999) Mechanical Interaction of Intersecting Normal Faults: Theory, Field Examples and Applications. Ph.D. Thesis, Stanford University, Stanford, CA.
  54. Martel, S.J. and Boger, W.A. (1998) Geometry and mechanics of secondary fracturing around small three‐dimensional faults in granitic rock. Journal of Geophysical Research: Solid Earth, 103(B9), 21299-21314.‏
  55. Kattenhorn, S.A., Aydin, A., and Pollard, D.D. (2000) Joints at high angles to normal fault strike: an explanation using 3-D numerical models of fault-perturbed stress fields. Journal of Structural Geology, 22(1), 1-23.‏
  56. Bourne, S.J. and Willemse, E.J. (2001) Elastic stress control on the pattern of tensile fracturing around a small fault network at Nash Point, UK. Journal of Structural Geology, 23(11), 1753-1770.‏
  57. Soliva, R., Maerten, F., Petit, J.P., and Auzias, V. (2010) Field evidences for the role of static friction on fracture orientation in extensional relays along strike-slip faults: comparison with photoelasticity and 3-D numerical modeling. Journal of Structural Geology, 32(11), 1721-1731.‏
  58. Bai, T., Maerten, L., Gross, M.R., and Aydin, A. (2002) Orthogonal cross joints: do they imply a regional stress rotation? Journal of Structural Geology, 24(1), 77-88.‏
  59. Crider, J.G. and Pollard, D.D. (1998) Fault linkage: Three‐dimensional mechanical interaction between echelon normal faults. Journal of Geophysical Research: Solid Earth, 103(B10), 24373-24391.
  60. Soliva, R., Benedicto, A., and Maerten, L. (2006) Spacing and linkage of confined normal faults: importance of mechanical thickness. Journal of Geophysical Research: Solid Earth, 111(B1).‏
  61. Savage, H.M. and Cooke, M.L. (2003) Can flat-ramp-flat fault geometry be inferred from fold shape?: A comparison of kinematic and mechanical folds. Journal of Structural Geology, 25(12), 2023-2034.‏
  62. Bellahsen, N., Fiore, P.E., and Pollard, D.D. (2006) From spatial variation of fracture patterns to fold kinematics: A geomechanical approach. Geophy-sical Research Letters, 33(2).‏
  63. White, I.R. and Crider, J.G. (2006) Extensional fault-propagation folds: Mechanical models and observations from the Modoc Plateau, northeastern California. Journal of Structural Geology, 28(7), 1352-1370.‏
  64. Allward-Fiore, P. (2006) 3D Characterization and Mechanics of Brittle Deformation in Thrust Fault Related Folds. D. Thesis, Stanford University, CA, USA.‏
  65. Mynatt, I., Hilley, G.E., and Pollard, D.D. (2007) Inferring fault characteristics using fold geometry constrained by Airborne Laser Swath Mapping at Raplee Ridge, Utah. Geophysical Research Letters, 34(16).‏
  66. Shackleton, J.R. and Cooke, M.L. (2007) Is plane strain a valid assumption in non-cylindrical fault-cored folds? Journal of Structural Geology, 29(7), 1229-1240.‏
  67. Hilley, G.E., Mynatt, I., and Pollard, D.D. (2010) Structural geometry of raplee ridge monocline and thrust fault imaged using inverse boundary element modeling and ALSM data. Journal of Structural Geology, 32(1), 45-58.‏
  68. Griffith, W.A. and Cooke, M.L. (2004) Mechanical validation of the three-dimensional intersection geometry between the Puente Hills blind-thrust system and the Whittier fault, Los Angeles, California. Bulletin of the Seismological Society of America, 94(2), 493-505.‏
  69. Olson, E.L. and Cooke, M.L. (2005) Application of three fault growth criteria to the Puente Hills thrust system, Los Angeles, California, USA. Journal of Structural Geology, 27(10), 1765-1777.‏
  70. Brankman, C.M. and Aydin, A. (2004) Uplift and contractional deformation along a segmented strike-slip fault system: the Gargano Promontory, southern Journal of Structural Geology, 26(5), 807-824.‏
  71. Resor, P.G. (2003) Deformation Associated with Continental Normal Faults. Doctoral dissertation, Stanford University.‏
  72. Grant, J.V. and Kattenhorn, S.A. (2004) Evolution of vertical faults at an extensional plate boundary, southwest Iceland. Journal of Structural Geology, 26(3), 537-557.‏
  73. Soliva, R., Benedicto, A., Schultz, R.A., Maerten, L., and Micarelli, L. (2008) Displacement and interaction of normal fault segments branched at depth: Implications for fault growth and potential earthquake rupture size. Journal of Structural Geology, 30(10), 1288-1299.‏
  74. Meigs, A.J., Cooke, M.L., and Marshall, S.T. (2008) Using vertical rock uplift patterns to constrain the three-dimensional fault configuration in the Los Angeles Basin. Bulletin of the Seismological Society of America, 98(1), 106-123.‏
  75. Maerten, F., Resor, P., Pollard, D., and Maerten, L. (2005) Inverting for slip on three-dimensional fault surfaces using angular dislocations. Bulletin of the Seismological Society of America, 95(5), 1654-1665.‏
  76. Johnson, K.M. and Segall, P. (2004) Imaging the ramp–décollement geometry of the Chelungpu fault using coseismic GPS displacements from the 1999 Chi-Chi, Taiwan earthquake. Tectonophysics, 378(1-2), 123-139.
  77. ‏Tselentis, G.A. and Gkika, F. (2005) Boundary element slope instability modeling of Corinth Canal, Greece due to nearby fault activation. WIT Transactions on the Built Environment, 78.
  78. Maerten, L., Gillespie, P., and Pollard, D.D. (2002) Effects of local stress perturbation on secondary fault development. Journal of Structural Geology, 24(1), 145-153.
  79. Childs, C., Watterson, J., and Walsh, J.J. (1995) Fault overlap zones within developing normal fault systems. Journal of the Geological Society, 152(3), 535-549.

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  • Receive Date: 13 April 2020
  • Revise Date: 22 June 2020
  • Accept Date: 03 February 2021

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