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Explosion Dynamics in Saturated Rocks and Solids


Non-elastic pore deformations and crack propagations are the principal causes of dynamic damage in rocks and soils. In the case of downhole blasting from wellbores, these two mechanisms compete with each other. Therefore, to carry out a mechanical analysis of rock blasting, a sufficiently complete model that takes these various mechanisms into account has to be developed. To address this issue, this paper proposes the use of an elastic–plastic model, which includes a yield condition with a non-associated plastic flow rule, the effects of pore fluid saturation, and a brittle failure criterion under extension. The results presented in this paper describe underground explosions with spherical motion (cavity growth under the internal pressure of detonated gases without leakage into the formation), typical for oil or water reservoirs. The governing equations are written in a Cartesian system of coordinates for the case of spatial dynamic medium deformation. For this case, Cartesian coordinates are more convenient than spherical coordinates because they avoid numerical difficulties connected with the non-divergent terms of the non-linear form of the Biot–Frenkel equations. The numerical method uses the Wilkins approach, which has been generalized for the model described in this paper. The dilatancy of the material during plastic deformation is neglected for simplicity. The numerical results show that, when using typical parameters for relatively “soft” porous skeleton, the plastic flow overcomes the brittle failure. An extension zone only appears near the cavity. The results also show the presence of the two Biot P-waves. The second Biot wave, however, is only seen in the case of an extremely high permeability rock. Furthermore, in the case of the first Biot wave, the saturating liquid and the solid skeleton particles are moving with different velocities in a 100 darcy rock and with the same velocity in a 0.01 darcy rock. Calculated radial particle velocities as a function of the scaled radius are close to measured velocities in rigid dense media but are larger than measured ones in clays. It is suggested that the difference is due to different levels of water saturation, assumed full saturation in the calculation, partial saturation in the experiments.

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  1. E.I. Andriankin V.P. Koryvov (1959) ArticleTitleShock wave in a varying compressed plastic medium Proc. Acad. Sci. USSR. 128 IssueID2 257–266

  2. M.A. Biot (1956) ArticleTitleTheory of propagation of elastic waves in a fluid-saturated porous solid. 1. low-frequency range J. Acoust. Soc. Am. 28 IssueID2 168–178 Occurrence Handle10.1121/1.1908239

  3. Bovt, A. N. and Nikolaevskiy, V. N.: 1981, Dilatancy and mechanics of underground explosion, Mech. Deformable Solid Bodies Moscow. VINITI, 14, 129–169 (in Russian).

  4. Chadwick, P., Cox, A. B. and Hopkins, H. G.: 1964, Mechanics of deep underground explosions, Phil. Trans. R. Soc. Lond., Ser. A. 256 (1070).

  5. S.H. Cho K. Kaneko (2004) ArticleTitleInfluence of the applied pressure waveform on the dynamic fracture process in rock Int. J. Rock Mech. Mining Sci. 41 771–784 Occurrence Handle10.1016/j.ijrmms.2004.02.006

  6. Cuderman, J. F.: 1982, Multiple fracturing experiments – propellant and borehole consideration, SPE/DOE 10845, 535–546.

  7. Finger, M., Li E., Helm, F., Heyes, B., Hornig, H., MacGaier, R., Kahara, M. and Hidry, M.: 1976, Effect of composition on detonation properties of explosions, in: Proceedings of the 6th Symposium on Detonation, Coronado, California.

  8. W.L. Fourney (1993) Mechanisms of rock fragmentation by blasting J. Hudson (Eds) Comprehensive Rock Engineering NumberInSeriesVol. 4 Pergamon Press Oxford 39–69

  9. Fourney W. L., Dick, R. D. and Young, C.: 1995, Response of oil shale on fragmentation by cylindrical charges, Rock Mechanics and Rock Engineering. Springer, Berlin, pp. 37–57.

  10. S.S. Grigoryan (1964) ArticleTitleTowards the solution of the problem of underground explosion in soft soils Sov. Appl. Math. Mech. (PMM) 28 IssueID6 1070–1082

  11. A.S. Kompaneez (1956) ArticleTitleShock waves in plastic compressed medium Proc. Acad. Sci. USSR 109 IssueID1 49–51

  12. Koryavov, V. P.: 1965, Zone and front of cracks in elastic body under action of pressure, J. Appl. Mech. Techn. Phys. (PMTF), (6), 87–95.

  13. A.L. Latter E.A. Martinelli E. Teller (1959) ArticleTitleSeismic scaling law for underground explosions Phys. Fluids 2 IssueID3 280–282 Occurrence Handle10.1063/1.1705923

  14. F. Lysmer R.L. Kuhlemeyer (1969) ArticleTitleFinite dynamic model for infinite media ASCE 95 IssueID11 859–877

  15. V.N. Nikolaevskiy (1967) ArticleTitleLinks between volume and shear plastic strains and shock waves in soft soils Proc. Acad. Sci. USSR 177 IssueID3 542–545

  16. V.N. Nikolaevskiy (1981) ArticleTitleLimit velocity of fracture front and dynamic strength of brittle solids Int. J. Eng. Sci. 19 IssueID1 41–56 Occurrence Handle10.1016/0020-7225(81)90048-3

  17. V.N. Nikolaevskiy (1996) Geomechanics and Fluidodynamics Kluwer Dordrecht

  18. V.N. Nikolaevskij K.S. Basniev A.T. Gorbunov G.A. Zotov (1970) Mechanics of saturated porous media Nedra Moscow

  19. V.N. Rodionov V.V. Adushkin V.N. Kostyuchenko V.N. Nikolaevskiy S. Romashov A. V.M. Zvetkov (1971) Mechanical Effect of Underground Explosion Nedra Moscow

  20. V. Schenk (1970) ArticleTitleStress waves produced by a spherical explosive source in gravel sandy soil Geofysikalni Sbornik XVIII IssueID326 257–293

  21. H.L. Selberg (1951) ArticleTitleTransient compression waves from spherical and cylindrical cavities Arkiv fur fisik Band 5 IssueID7 97–108

  22. I.D. Sharpe (1942) ArticleTitleThe production od elastic waves by explosion pressure Geophysics 7 IssueID2 144–154 Occurrence Handle10.1190/1.1445002

  23. M.L. Wilkins (1999) Computer Simulation of Dynamic Phenomena Springer Berlin

  24. B.V. Zamyshljaev L.S. Evterev (1990) Models of Dynamic Deformation and Failure of Earth media Nauka Moscow

  25. A. G. Zhilenkov S. M. Kapustyanskiy V. N. Nikolaevskiy (2004) ArticleTitleEffects of limit failure velocity at dynamic expansion of cavity in brittle materials Mech. Solids 1 200–208

  26. A.I. Zhmakin A.A. Fusenko (1980) ArticleTitleOn one monotonous difference scheme of through calculation J. Comput. Math. Math. Phys. 20 IssueID4 1021–1031

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Correspondence to M. Thiercelin.

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Nikolaevskiy, V.N., Kapustyanskiy, S.M., Thiercelin, M. et al. Explosion Dynamics in Saturated Rocks and Solids. Transp Porous Med 65, 485–504 (2006).

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  • dynamics
  • elasto-plasticity
  • saturation
  • soils
  • rocks
  • explosion
  • waves
  • permeability