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Introduction

  • Merab Svanadze
Chapter
Part of the Interdisciplinary Applied Mathematics book series (IAM, volume 51)

Abstract

This chapter is divided into seven main sections. In Sect. 1.1, a brief review of the theories of multi-porosity materials is presented. In Sect. 1.2, the short history of the potential method is introduced. In Sect. 1.3, the basic notations are given. These notations are used throughout this work. In Sects. 1.4 and 1.5, the basic equations of thermoelasticity and elasticity of quadruple porosity solids are presented, respectively. In Sect. 1.6, these equations are rewritten in the matrix form. Finally, in Sect. 1.7, the stress operators of the considered theories are given.

References

  1. 1.
    Abdassah, D., Ershaghi, I.: Triple-porosity systems for representing naturally fractured Reservoirs. SPE Form Eval. (April) 113–127. SPE-13409-PA (1986)Google Scholar
  2. 3.
    Aguilera, R.: Naturally Fractured Reservoirs, 2nd edn. PennWell Books, Tulsa (1995)Google Scholar
  3. 4.
    Aguilera, R.F., Aguilera, R.: A triple - porosity model for petrophysical analysis of naturally fractured reservoirs. Petrophysics 45, 157–166 (2004)Google Scholar
  4. 5.
    Aguilera, R., Lopez, B.: Evaluation of quintuple porosity in shale petroleum reservoirs. SPE Eastern Regional Meeting, 20–22 August, Pittsburgh. SPE-165681-MS, 28pp. (2013). https://doi.org/10.2118/165681-MS
  5. 6.
    Aifantis, E.C.: Introducing a multi-porous media. Dev. Mech. 9, 209–211 (1977)Google Scholar
  6. 7.
    Aifantis, E.C.: Further comments on the problem of heat extraction from hot dry rocks. Mech. Res. Commun. 7, 219–226 (1980)CrossRefGoogle Scholar
  7. 8.
    Aifantis, E.C., Beskos, D.E.: Heat extraction from hot dry rocks. Mech. Res. Commun. 7, 165–170 (1980)MathSciNetCrossRefGoogle Scholar
  8. 13.
    Aouadi, M.: A theory of thermoelastic diffusion materials with voids. Z. Angew. Math. Phys. 61, 357–379 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 14.
    Aouadi, M.: Uniqueness and existence theorems in thermoelasticity with voids without energy dissipation. J. Franklin Inst. 349, 128–139 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 15.
    Aouadi, M.: Stability in thermoelastic diffusion theory with voids. Appl. Anal. 91, 121–139 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 16.
    Arbogast, T., Douglas, J., Hornung, U.: Derivation of the double porosity model of single phase flow via homogenization theory. SIAM J. Math. Anal. 21, 823–836 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 17.
    Arusoaie, A.: Spatial and temporal behavior in the theory of thermoelasticity for solids with double porosity. J. Therm. Stresses 41, 500–521 (2018)CrossRefGoogle Scholar
  13. 19.
    Ba, J., Carcione, J.M., Nie, J.X.: Biot-Rayleigh theory of wave propagation in double-porosity media. J. Geophys. Res. 116, B06202 (2011). https://doi.org/10.1029/2010JB008185 CrossRefGoogle Scholar
  14. 20.
    Bai, M., Roegiers, J.C.: Fluid flow and heat flow in deformable fractured porous media. Int. J. Eng. Sci. 32, 1615–1633 (1994)zbMATHCrossRefGoogle Scholar
  15. 21.
    Bai, M., Roegiers, J.C.: Triple-porosity analysis of solute transport. J. Cantam. Hydrol. 28, 189–211 (1997)Google Scholar
  16. 22.
    Bai, M., Elsworth, D., Roegiers, J.C.: Multiporosity/multipermeability approach to the simulation of naturally fractured reservoirs. Water Resour. Res. 29, 1621–1633 (1993)CrossRefGoogle Scholar
  17. 24.
    Barenblatt, G.I., Zheltov, Y.P.: Fundamental equations of filtration of homogeneous liquids in fissured rock. Sov. Phys. Dokl. 5, 522–525 (1960)zbMATHGoogle Scholar
  18. 25.
    Barenblatt, G.I., Zheltov, Y.P., Kochina, I.N.: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks (strata). J. Appl. Math. Mech. 24, 1286–1303 (1960)zbMATHCrossRefGoogle Scholar
  19. 31.
    Basheleishvili, M., Bitsadze, L.: Explicit solutions of the boundary value problems of the theory of consolidation with double porosity for the half-plane. Georgian Math. J. 19, 41–48 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 35.
    Bazarra, N., Fernández, J.R., Leseduarte, M.C., Magaña, A., Quintanilla, R.: On the thermoelasticity with two porosities: asymptotic behaviour. Math. Mech. Solids (2018, in press). 24, 2713–2725 (2019)Google Scholar
  21. 36.
    Bear, J.: Modeling Phenomena of Flow and Transport in Porous Media. Springer, Basel (2018)zbMATHCrossRefGoogle Scholar
  22. 37.
    Berryman, J.G., Wang, H.F.: The elastic coefficients of double - porosity models for fluid transport in jointed rock. J. Geophys. Res. B 100, 24611–24627 (1995)CrossRefGoogle Scholar
  23. 38.
    Berryman, J.G., Wang, H.F.: Elastic wave propagation and attenuation in a double-porosity dual-permeability medium. Int. J. Rock Mech. Min. Sci. 37, 63–78 (2000)CrossRefGoogle Scholar
  24. 40.
    Beskos, D.E., Aifantis, E.C.: On the theory of consolidation with double porosity - II. Int. J. Eng. Sci. 24, 1697–1716 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 41.
    Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941)zbMATHCrossRefGoogle Scholar
  26. 44.
    Biot, M.A.: Thermoelasticity and irreversible thermodynamics. J. Appl. Phys. 27, 240–253 (1956)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 48.
    Biot, M.A.: Variational Lagrangian-thermodynamics of nonisothermal finite strain mechanics of porous solids and thermomolecular diffusion. Int. J. Solids Struct. 13, 579–597 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 56.
    Bitsadze, L., Tsagareli, I.: Solutions of BVPs in the fully coupled theory of elasticity for the space with double porosity and spherical cavity. Math. Methods Appl. Sci. 39, 2136–2145 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 57.
    Bitsadze, L., Tsagareli, I.: The solution of the Dirichlet BVP in the fully coupled theory of elasticity for spherical layer with double porosity. Meccanica 51, 1457–1463 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 59.
    Bluhm, J., de Boer, R.: The volume fraction concept in the porous media theory. ZAMM J. Appl. Math. Mech. 77, 563–577 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 60.
    Bowen, R.M.: Incompressible porous media models by use of the theory of mixtures. Int. J. Eng. Sci. 18, 1129–1148 (1980)zbMATHCrossRefGoogle Scholar
  32. 61.
    Bowen, R.M.: Compressible porous media models by use of the theory of mixtures. Int. J. Eng. Sci. 20, 697–735 (1982)zbMATHCrossRefGoogle Scholar
  33. 64.
    Burchuladze, T.V., Gegelia, T.G.: The Development of the Potential Methods in the Elasticity Theory (Russian). Metsniereba, Tbilisi (1985)Google Scholar
  34. 65.
    Burchuladze, T., Svanadze, M.: Potential method in the linear theory of binary mixtures for thermoelastic solids. J. Therm. Stresses 23, 601–626 (2000)MathSciNetCrossRefGoogle Scholar
  35. 68.
    Casas, P.S., Quintanilla, R.: Exponential decay in one-dimensional porous-thermo-elasticity. Mech. Res. Commun. 32, 652–658 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 78.
    Cheng, A.H.D.: Poroelasticity. Springer, Basel (2016)zbMATHCrossRefGoogle Scholar
  37. 79.
    Cheng, A.H.D., Cheng, D.T.: Heritage and early history of the boundary element method. Eng. Anal. Bound. Elem. 29, 268–302 (2005)zbMATHCrossRefGoogle Scholar
  38. 80.
    Chirita, S.: Rayleigh waves on an exponentially graded poroelastic half space. J. Elast. 110, 185–199 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 84.
    Chirita, S., Ghiba, I.D.: Strong ellipticity and progressive waves in elastic materials with voids. Proc. Roy. Soc. Lond. A 466, 439–458 (2010)zbMATHCrossRefGoogle Scholar
  40. 85.
    Chirita, S., Scalia, A.: On the spatial and temporal behavior in linear thermoelasticity of materials with voids. J. Therm. Stresses 24, 433–455 (2001)MathSciNetCrossRefGoogle Scholar
  41. 87.
    Chirita, S., Ciarletta, M., Straughan, B.: Structural stability in porous elasticity. Proc. Roy. Soc. Lond. A 462, 2593–2605 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 89.
    Ciarletta, M., Ieşan, D.: Non-Classical Elastic Solids. Longman Scientific and Technical. Wiley, New York (1993)Google Scholar
  43. 90.
    Ciarletta, M., Chirita, S., Passarella, F.: Some results on the spatial behaviour in linear porous elasticity. Arch. Mech. 57, 43–65 (2005)MathSciNetzbMATHGoogle Scholar
  44. 94.
    Ciarletta, M., Straughan, B.: Thermo-poroacoustic acceleration waves in elastic materials with voids. J. Math. Anal. Appl. 333, 142–150 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 95.
    Ciarletta, M., Scalia, A., Svanadze, M.: Fundamental solution in the theory of micropolar thermoelasticity for materials with voids. J. Therm. Stresses 30, 213–229 (2007)MathSciNetCrossRefGoogle Scholar
  46. 97.
    Ciarletta, M., Svanadze, M., Buonano, L.: Plane waves and vibrations in the micropolar thermoelastic materials with voids. Eur. J. Mech. A Solids 28, 897–903 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 99.
    Ciarletta, M., Passarella, F., Svanadze, M.: Plane waves and uniqueness theorems in the coupled linear theory of elasticity for solids with double porosity. J. Elast. 114, 55–68 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 100.
    Ciarletta, M., Straughan, B., Tibullo, V.: Acceleration waves in a nonlinear Biot theory of porous media. Int. J. Non-Linear Mech. 103, 23–26 (2018)CrossRefGoogle Scholar
  49. 106.
    Coussy, O.: Poromechanics. Wiley, Chichester (2004)zbMATHGoogle Scholar
  50. 107.
    Coussy, O.: Mechanics and Physics of Porous Solids. Wiley, Chichester (2010)CrossRefGoogle Scholar
  51. 108.
    Cowin, S.C.: The stresses around a hole in a linear elastic material with voids. Quart. J. Mech. Appl. Math. 37, 441–465 (1984)zbMATHCrossRefGoogle Scholar
  52. 110.
    Cowin, S.C.: Bone poroelasticity. J. Biomech. 32, 217–238 (1999)CrossRefGoogle Scholar
  53. 111.
    Cowin, S.C. (ed.): Bone Mechanics Handbook. Informa Healthcare USA, New York (2008)Google Scholar
  54. 112.
    Cowin, S.C., Nunziato, J.W.: Linear elastic materials with voids. J. Elast. 13, 125–147 (1983)zbMATHCrossRefGoogle Scholar
  55. 113.
    Cowin, S.C., Puri, P.: The classical pressure vessel problems for linear elastic materials with voids. J. Elast. 13, 157–163 (1983)zbMATHCrossRefGoogle Scholar
  56. 114.
    Cushman, J.H.: The Physics of Fluids in Hierarchical Porous Media: Angstroms to Miles. Springer, Dordrecht (1997)CrossRefGoogle Scholar
  57. 116.
    Dai, W.Z., Kuang, Z.B.: Love waves in double porosity media. J. Sound Vib. 296, 1000–1012 (2006)CrossRefGoogle Scholar
  58. 117.
    Dai, Z.J., Kuang, Z.B., Zhao, S.X.: Rayleigh waves in a double porosity half-space. J. Sound Vib. 298, 319–332 (2006)CrossRefGoogle Scholar
  59. 118.
    Dai, Z.J., Kuang, Z.B., Zhao, S.X.: Reflection and transmission of elastic waves from the interface of a fluid-saturated porous solid and a double porosity solid. Transp. Porous Media 65, 237–264 (2006)MathSciNetCrossRefGoogle Scholar
  60. 120.
    D’Apice, C., Chirita, S.: Plane harmonic waves in the theory of thermoviscoelastic materials with voids. J. Therm. Stresses 39, 142–155 (2016)CrossRefGoogle Scholar
  61. 122.
    Das, M.K., Mukherjee, P.P., Muralidhar, K.: Modeling Transport Phenomena in Porous Media with Applications. Springer, Cham (2018)zbMATHCrossRefGoogle Scholar
  62. 123.
    de Boer, R.: Theory of Porous Media: Highlights in the Historical Development and Current State. Springer, Berlin (2000)zbMATHCrossRefGoogle Scholar
  63. 124.
    de Boer, R.: Contemporary progress in porous media theory. Appl. Mech. Rev. 53, 323–370 (2000)CrossRefGoogle Scholar
  64. 125.
    de Boer, R.: Trends in Continuum Mechanics of Porous Media. Springer, Dordrecht (2005)zbMATHCrossRefGoogle Scholar
  65. 129.
    de Boer, R., Svanadze, M.: Fundamental solution of the system of equations of steady oscillations in the theory of fluid-saturated porous media. Transp. Porous Media 56, 39–50 (2004)MathSciNetCrossRefGoogle Scholar
  66. 131.
    De Cicco, S., Diaco, M.: A theory of thermoelastic materials with voids without energy dissipation. J. Therm. Stresses 25, 493–503 (2002)MathSciNetCrossRefGoogle Scholar
  67. 132.
    Dhaliwal, R.S., Wang, J.: A heat-flux dependent theory of thermoelasticity with voids. Acta Mech. 110, 33–39 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  68. 133.
    Dormieux, L., Kondo, D., Ulm, F.-J.: Microporomechanics. Wiley, Chichester (2006)zbMATHCrossRefGoogle Scholar
  69. 139.
    Florea, O.: Spatial behavior in thermoelastodynamics with double porosity structure. Int. J. Appl. Mech. 9, 1750097, 14pp. (2017). https://doi.org/10.1142/S1758825117500971
  70. 140.
    Florea, O.A.: Harmonic vibrations in thermoelastic dynamics with double porosity structure. Math. Mech. Solids (2018). 24, 2410–2424 (2019)Google Scholar
  71. 142.
    Franchi, F., Lazzari, B., Nibbi, R., Straughan, B.: Uniqueness and decay in local thermal non-equilibrium double porosity thermoelasticity. Math. Methods Appl. Sci. 41, 6763–6771 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  72. 144.
    Fredholm, I.: Sur une classe d’équations fonctionelles. Acta Math. 27, 365–390 (1903)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 147.
    Gegelia, T., Jentsch, L.: Potential methods in continuum mechanics. Georgian Math. J. 1, 599–640 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  74. 149.
    Gelet, R., Loret, B., Khalili, N.: Borehole stability analysis in a thermoporoelastic dual-porosity medium. Int. J. Rock Mech. Min. Sci. 50, 65–76 (2012)CrossRefGoogle Scholar
  75. 152.
    Gentile, M., Straughan, B.: Acceleration waves in nonlinear double porosity elasticity. Int. J. Eng. Sci. 73, 10–16 (2013)zbMATHCrossRefGoogle Scholar
  76. 157.
    Green, G.: An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. Nottingham (1828)Google Scholar
  77. 161.
    Günther, N.M.: Potential Theory and its Applications to Basic Problems of Mathematical Physics. Ungar, New York (1967)Google Scholar
  78. 164.
    He, J., Teng, W., Xu, J., Jiang, R., Sun, J.: A quadruple-porosity model for shale gas reservoirs with multiple migration mechanisms. J. Nat. Gas Sci. Eng. 33, 918–933 (2016)CrossRefGoogle Scholar
  79. 166.
    Holzapfel, G.A., Ogden, R.W. (eds): Biomechanics: Trends in Modeling and Simulation. Springer, Basel (2017)zbMATHGoogle Scholar
  80. 169.
    Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations. Springer, Berlin (2008)zbMATHCrossRefGoogle Scholar
  81. 171.
    Ichikawa, Y., Selvadurai, A.P.S.: Transport Phenomena in Porous Media: Aspects of Micro/Macro Behaviour. Springer, Berlin (2012)CrossRefGoogle Scholar
  82. 172.
    Ieşan, D.: Shock waves in micropolar elastic materials with voids. An. St. Univ. Al. I. Cuza Iasi. 81, 177–186 (1985)MathSciNetzbMATHGoogle Scholar
  83. 173.
    Ieşan, D.: Some theorems in the theory of elastic materials with voids. J. Elast. 15, 215–224 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  84. 174.
    Ieşan, D.: A theory of thermoelastic materials with voids. Acta Mech. 60, 67–89 (1986)CrossRefGoogle Scholar
  85. 175.
    Ieşan, D.: Thermoelastic Models of Continua. Springer, Dordrecht (2004)zbMATHCrossRefGoogle Scholar
  86. 176.
    Ieşan, D.: Classical and Generalized Models of Elastic Rods. Chapman and Hall/CRC, New York (2008)zbMATHCrossRefGoogle Scholar
  87. 177.
    Ieşan, D.: On a theory of thermoviscoelastic materials with voids. J. Elast. 104, 369–384 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  88. 178.
    Ieşan, D.: Method of potentials in elastostatics of solids with double porosity. Int. J. Eng. Sci. 88, 118–127 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  89. 179.
    Ieşan, D.: On the prestressed thermoelastic porous materials. J. Therm. Stresses 41, 1212–1224 (2018)CrossRefGoogle Scholar
  90. 182.
    Ieşan, D., Quintanilla, R.: Non-linear deformations of porous elastic solids. Int. J. Non-Linear Mech. 49, 57–65 (2013)CrossRefGoogle Scholar
  91. 183.
    Ieşan, D., Quintanilla, R.: On a theory of thermoelastic materials with a double porosity structure. J. Therm. Stresses 37, 1017–1036 (2014)CrossRefGoogle Scholar
  92. 188.
    Janjgava, R.: Elastic equilibrium of porous Cosserat media with double porosity. Adv. Math. Phys. 2016, 4792148, 9pp. (2016). https://doi.org/10.1155/2016/4792148
  93. 192.
    Kansal, T.: Generalized theory of thermoelastic diffusion with double porosity. Arch. Mech. 70, 241–268 (2018)MathSciNetzbMATHGoogle Scholar
  94. 193.
    Kansal, T.: Fundamental solution of the system of equations of pseudo oscillations in the theory of thermoelastic diffusion materials with double porosity. Multidisc. Model. Mater. Struct. 20pp. (2018).  https://doi.org/10.1108/MMMS-01-2018-0006
  95. 195.
    Kellogg, O.D.: Foundations of Potential Theory. Springer, Berlin (1929)zbMATHCrossRefGoogle Scholar
  96. 197.
    Khaled, M.Y., Beskos, D.E., Aifantis, E.C.: On the theory of consolidation with double porosity - III: a finite element formulation. Nimer. Anal. Meth. Geomech. 8, 101–123 (1984)zbMATHCrossRefGoogle Scholar
  97. 198.
    Khalili, N.: Coupling effects in double porosity media with deformable matrix. Geophys. Res. Lett. 30, 22 (2003). https://doi.org/10.1029/2003GL018544 Google Scholar
  98. 199.
    Khalili, N., Habte, M.A., Zargarbashi, S.: A fully coupled flow deformation model for cyclic analysis of unsaturated soils including hydraulic and mechanical hysteresis. Comput. Geotech. 35, 872–889 (2008)CrossRefGoogle Scholar
  99. 200.
    Khalili, N., Selvadurai, A.P.S.: A fully coupled constitutive model for thermo-hydro-mechanical analysis in elastic media with double porosity. Geophys. Res. Lett. 30(24), 2268 (2003). https://doi.org/10.1029/2003GL018838 CrossRefGoogle Scholar
  100. 206.
    Kumar, R., Vohra, R.: State space approach to plane deformation in elastic material with double porosity. Mater. Phys. Mech. 24, 9–17 (2015)Google Scholar
  101. 207.
    Kumar, R., Vohra, R.: A problem of spherical cavity in an infinite generalized thermoelastic medium with double porosity subjected to moving heat source. Med. J. Model. Simul. 6, 67–81 (2016)Google Scholar
  102. 208.
    Kumar, R.M., Vohra, R.: Elastodynamic problem for an infinite body having a spherical cavity in the theory of thermoelasticity with double porosity. Mech. Mech. Eng. 21, 267–289 (2017)Google Scholar
  103. 209.
    Kumar, R.M., Vohra, R.: Vibration analysis of thermoelastic double porous microbeam subjected to laser pulse. Mech. Adv. Mater. Struct. 26, 471–479 (2017). https://doi.org/10.1080/15376494.2017.1341578 CrossRefGoogle Scholar
  104. 210.
    Kumar, R., Vohra, R., Gorla, M.G.: Some considerations of fundamental solution in micropolar thermoelastic materials with double porosity. Arch. Mech. 68, 263–284 (2016)MathSciNetzbMATHGoogle Scholar
  105. 211.
    Kumar, R., Vohra, R., Gorla, M.G.: Thermomechanical response in thermoelastic medium with double porosity. J. Solid Mech. 9, 24–38 (2017)Google Scholar
  106. 212.
    Kumar, R., Vohra, R., Gorla, M.G.: Variational principle and plane wave propagation in thermoelastic medium with double porosity under Lord-Shulman theory. J. Solid Mech. 9, 423–433 (2017)Google Scholar
  107. 213.
    Kupradze, V.D.: The existence and uniqueness theorems in the diffraction theory (Russian). Doklady AN SSSR 1, 235–240 (1934)zbMATHGoogle Scholar
  108. 214.
    Kupradze, V.D.: Solution of boundary value problems of Helmholtz equations in extraordinary cases (Russian). Doklady AN SSSR 1, 521–526 (1934)Google Scholar
  109. 215.
    Kupradze, V.D.: Boundary Value Problems of the Oscillation Theory and Integral Equations. M.-L., State Publishing House of technical and theoretical Literature (1950) (Russian). German translation: Kupradze, V.D.: Randwertaufgaben der Schwingungstheorie und Integralgleichongen. Veb Deutscher verlag der Wissenschaften, Berlin (1956)Google Scholar
  110. 216.
    Kupradze, V.D.: Dynamical problems in elasticity. In: Sneddon, I.N., Hill, R. (eds.) Progress in Solid Mechanics, vol. III, pp. 1–259. North Holland, Amsterdam (1963)Google Scholar
  111. 217.
    Kupradze, V.D.: Potential Methods in the Theory of Elasticity. Israel Program for Scientific Translations, Jerusalem (1965)Google Scholar
  112. 219.
    Kupradze, V.D., Gegelia, T.G., Basheleishvili, M.O., Burchuladze, T.V.: Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity. North-Holland, Amsterdam (1979)Google Scholar
  113. 225.
    Liu, C., Abousleiman, Y.N.: N-porosity and N-permeability generalized wellbore stability analytical solutions and applications: 50th US Rock Mechanics/Geomechanics Symposium, ARMA, 16-417, 9pp. (2016)Google Scholar
  114. 226.
    Liu, C.Q.: Exact solution for the compressible flow equations through a medium with triple-porosity. Appl. Math. Mech. 2, 457–462 (1981)zbMATHCrossRefGoogle Scholar
  115. 227.
    Liu, J.C., Bodvarsson, G.S., Wu, Y.S.: Analysis of pressure behaviour in fractured lithophysical reservoirs. J. Cantam. Hydrol. 62–63, 189–211 (2003)CrossRefGoogle Scholar
  116. 228.
    Liu, Z.: Multiphysics in Porous Materials. Springer, Basel (2018)CrossRefGoogle Scholar
  117. 232.
    Lopez, B., Aguilera, R.: Physics-based approach for shale gas numerical simulation: quintuple porosity and gas diffusion from solid kerogen. In: Presented at the SPE Annual Technical Conference and Exhibition, Houston, 28–30 September. SPE-175115-MS, 32pp. (2015). https://doi.org/10.2118/175115-MS
  118. 233.
    Lopez, B., Aguilera, R.: Petrophysical quantification of multiple porosities in shale-petroleum reservoirs with the use of modified pickett plots. SPE Reserv. Eval. Eng. SPE-171638-PA, 15pp. (2017). https://doi.org/10.2118/171638-PA
  119. 235.
    Magaña, A., Quintanilla, R.: On the time decay of solutions in one-dimensional theories of porous materials. Int. J. Solids Struct. 43, 3414–3427 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  120. 239.
    Marin, M.: Some basic theorems in elastostatics of micropolar materials with voids. J. Comput. Appl. Math. 70, 115–126 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  121. 241.
    Marin, M., Nicaise, S.: Existence and stability results for thermoelastic dipolar bodies with double porosity. Cont. Mech. Thermodynam. 28, 1645–1657 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  122. 242.
    Marin, M., Vlase, S., Paun, M.: Considerations on double porosity structure for micropolar bodies. AIP Adv. 5, 037113, 10pp. (2015). https://doi.org/10.1063/1.4914912
  123. 243.
    Masters, I., Pao, W.K.S., Lewis, R.W.: Coupling temperature to a double - porosity model of deformable porous media. Int. J. Numer. Methods Eng. 49, 421–438 (2000)zbMATHCrossRefGoogle Scholar
  124. 245.
    Mehrabian, A.: The poroelastic constants of multiple-porosity solids. Int. J. Eng. Sci. 132, 97–104 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  125. 246.
    Mehrabian, A., Abousleiman, Y.N.: Generalized Biot’s theory and Mandel’s problem of multiple-porosity and multiple-permeability poroelasticity. J. Geophys. Res. Solid Earth. 119, 2745–2763 (2014)CrossRefGoogle Scholar
  126. 247.
    Mehrabian, A., Abousleiman, Y.N.: Multiple-porosity and multiple-permeability poroelasticity: theory and benchmark analytical solution. In: Vandamme, M., Dangla, P., Pereira, J.M., Siavash Ghabezloo, S. (eds.) Poromechanics VI: Proceedings of the Sixth Biot Conference on Poromechanics, pp. 262–271 (2017). https://doi.org/10.1061/9780784480779.032
  127. 248.
    Mehrabian, A., Abousleiman, Y.N.: Theory and analytical solution to Cryer’s problem of N-porosity and N-permeability poroelasticity. J. Mech. Phys. Solids 118, 218–227 (2018)MathSciNetCrossRefGoogle Scholar
  128. 252.
    Mikhlin, S.G.: Multidimensional Singular Integrals and Integral Equations. Pergamon Press, Oxford (1965)zbMATHGoogle Scholar
  129. 254.
    Moutsopoulos, K.N., Konstantinidis, A.A., Meladiotis, I., Tzimopoulos, C.D., Aifantis, E.C.: Hydraulic behavior and contaminant transport in multiple porosity media. Transp. Porous Media 42, 265–292 (2001)CrossRefGoogle Scholar
  130. 256.
    Muskhelishvili, N.I.: Singular Integral Equations. Noordhoff, Groningen (1953)zbMATHGoogle Scholar
  131. 257.
    Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen (1953)zbMATHGoogle Scholar
  132. 261.
    Nield, D.A., Bejan, A.: Convection in Porous Media, 5th edn. Springer, Basel (2017)zbMATHCrossRefGoogle Scholar
  133. 262.
    Nikolaevskij, V.N.: Mechanics of Porous and Fractured Media. World Scientific, Singapore (1990)zbMATHCrossRefGoogle Scholar
  134. 266.
    Nunziato, J.W., Cowin, S.C.: A nonlinear theory of elastic materials with voids. Arch. Rat. Mech. Anal. 72, 175–201 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  135. 272.
    Passarella, F.: Some results in micropolar thermoelasticity. Mech. Res. Commun. 23, 349–357 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  136. 274.
    Passarella, F., Tibullo, V., Zampoli, V.: On the heat-flux dependent thermoelasticity for micropolar porous media. J. Therm. Stresses 34, 778–794 (2011)zbMATHCrossRefGoogle Scholar
  137. 275.
    Patwardhan, S.D., Famoori, F., Govindarajan, S.K.: Quad-porosity shale systems - a review. World J. Eng. 13, 529–539 (2016)CrossRefGoogle Scholar
  138. 278.
    Pride, S.R., Berryman, J.G.: Linear dynamics of double-porosity dual-permeability materials I. Governing equations and acoustic attenuation. Phys. Rev. E 68, 036603 (2003)Google Scholar
  139. 279.
    Puri, P., Cowin, S.C.: Plane waves in linear elastic material with voids. J. Elast. 15, 167–183 (1985)zbMATHCrossRefGoogle Scholar
  140. 282.
    Quintanilla, R.: Slow decay for one-dimensional porous dissipation elasticity. Appl. Math. Lett. 16, 487–491 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  141. 285.
    Radhika, B.P., Krishnamoorthy, A., Rao, A.U.: A review on consolidation theories and its application. Int. J. Geotech. Eng. 8pp. (2017). https://doi.org/10.1080/19386362.2017.1390899
  142. 288.
    Rezaee, R.: Fundamentals of Gas Shale Reservoirs. Wiley, Hoboken (2015)CrossRefGoogle Scholar
  143. 289.
    Ricken, T., Bluhm, J.: Remodeling and growth of living tissue: a multiphase theory. Arch. Appl. Mech. 80, 453–465 (2010)zbMATHCrossRefGoogle Scholar
  144. 294.
    Sang, G., Elsworth, D., Miao, X., Mao, X., Wang, J.: Numerical study of a stress dependent triple porosity model for shale gas reservoirs accommodating gas diffusion in kerogen. J. Nat. Gas Sci. Eng. 32, 423–438 (2016)CrossRefGoogle Scholar
  145. 297.
    Scalia, A.: Harmonic oscillations of a rigid punch on a porous elastic layer. J. Appl. Math. Mech. 73, 344–350 (2009)zbMATHCrossRefGoogle Scholar
  146. 299.
    Scalia, A., Svanadze, M.: Potential method in the linear theory of thermoelasticity with microtemperatures. J. Therm. Stresses 32, 1024–1042 (2009)CrossRefGoogle Scholar
  147. 300.
    Scalia, A., Svanadze, M.: Basic theorems in thermoelastostatics of bodies with microtemperatures. In: Hetnarski, R.B.(ed), Encyclopedia of Thermal Stresses, 11 vols, pp. 355–365, 1st edn. Springer, Berlin (2014)Google Scholar
  148. 301.
    Scalia, A., Svanadze, M., Tracinà, R.: Basic theorems in the equilibrium theory of thermoelasticity with microtemperatures. J. Therm. Stresses 33, 721–753 (2010)CrossRefGoogle Scholar
  149. 302.
    Scarpetta, E.: On the fundamental solutions in micropolar elasticity with voids. Acta Mech. 82, 151–158 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  150. 304.
    Scarpetta, E., Svanadze, M.: Uniqueness theorems in the quasi-static theory of thermoelasticity for solids with double porosity, J. Elast. 120, 67–86 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  151. 305.
    Scarpetta, E., Svanadze, M., Zampoli, V.: Fundamental solutions in the theory of thermoelasticity for solids with double porosity. J. Therm. Stresses 37, 727–748 (2014)CrossRefGoogle Scholar
  152. 307.
    Selvadurai, A.P.S.: The analytical method in geomechanics. Appl. Mech. Rev. ASME 60, 87–106 (2007)CrossRefGoogle Scholar
  153. 308.
    Selvadurai, A.P.S., Suvorov, A.: Thermo-Poroelasticity and Geomechanics. Cambridge University Press, Cambridge (2017)CrossRefGoogle Scholar
  154. 310.
    Sheng, G., Su, Y., Wang, W., Liu, J., Lu, M., Zhang, Q., Ren, L.: A multiple porosity media model for multi-fractured horizontal wells in shale gas reservoirs. J. Nat. Gas Sci. Eng. 27, 1562–1573 (2015)CrossRefGoogle Scholar
  155. 311.
    Showalter, R.E., Visarraga, D.B.: Double-diffusion models from a highly heterogeneous medium. J. Math. Anal. Appl. 295, 191–210 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  156. 312.
    Showalter, R.E., Walkington, N.J.: Micro-structure models of diffusion in fissured media. J. Math. Anal. Appl. 155, 1–20 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  157. 319.
    Straughan, B.: Stability and Wave Motion in Porous Media. Springer, New York (2008)zbMATHGoogle Scholar
  158. 320.
    Straughan, B.: Heat Waves. Applied Mathematical Sciences, vol. 177. Springer, New York (2011)Google Scholar
  159. 322.
    Straughan, B.: Stability and uniqueness in double porosity elasticity. Int. J. Eng. Sci. 65, 1–8 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  160. 323.
    Straughan, B.: Convection with Local Thermal Non-Equilibrium and Microfluidic Effects. Springer, Berlin (2015)zbMATHCrossRefGoogle Scholar
  161. 324.
    Straughan, B.: Modelling questions in multi-porosity elasticity. Meccanica 51, 2957–2966 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  162. 326.
    Straughan, B.: Mathematical Aspects of Multi-Porosity Continua. Springer, Basel (2017)zbMATHCrossRefGoogle Scholar
  163. 327.
    Straughan, B.: Solid mechanics–uniqueness and stability in triple porosity thermoelasticity. Rend. Lincei Mat. Appl. 28, 191–208 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  164. 329.
    Su, B.-L., Sanchez, C., Yang, X.-Y. (eds): Hierarchically Structured Porous Materials: From Nanoscience to Catalysis, Separation, Optics, Energy, and Life Science. Wiley-VCH Verlag, Weinheim (2012)Google Scholar
  165. 330.
    Svanadze, M.: On existence of eigenfrequencies in the theory of two-component elastic mixtures. Quart. J. Mech. Appl. Math. 51, 427–437 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  166. 331.
    Svanadze, M.: Fundamental solution in the theory of consolidation with double porosity. J. Mech. Behav. Mater. 16(1–2), 123–130 (2005)Google Scholar
  167. 332.
    Svanadze, M.: Plane waves and eigenfrequencies in the linear theory of binary mixtures of thermoelastic solids. J. Elast. 92, 195–207 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  168. 333.
    Svanadze, M.: Dynamical problems of the theory of elasticity for solids with double porosity. Proc. Appl. Math. Mech. 10(1), 309–310 (2010)MathSciNetCrossRefGoogle Scholar
  169. 334.
    Svanadze, M.: Plane waves and boundary value problems in the theory of elasticity for solids with double porosity. Acta Appl. Math. 122, 461–471 (2012)MathSciNetzbMATHGoogle Scholar
  170. 335.
    Svanadze, M.: The boundary value problems of the fully coupled theory of poroelasticity for materials with double porosity. Proc. Appl. Math. Mech. 12(1), 279–282 (2012)CrossRefGoogle Scholar
  171. 336.
    Svanadze, M.: Fundamental solution in the linear theory of consolidation for elastic solids with double porosity. J. Math. Sci. 195, 258–268 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  172. 337.
    Svanadze, M.: Uniqueness theorems in the theory of thermoelasticity for solids with double porosity. Meccanica 49, 2099–2108 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  173. 338.
    Svanadze, M.: On the theory of viscoelasticity for materials with double porosity. Disc. Contin. Dynam. Syst. Ser. B 19, 2335–2352 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  174. 339.
    Svanadze, M.: Boundary value problems in the theory of thermoporoelasticity for materials with double porosity. Proc. Appl. Math. Mech. 14(1), 327–328 (2014)MathSciNetCrossRefGoogle Scholar
  175. 340.
    Svanadze, M.: Large existence of solutions in thermoelasticity theory of steady vibrations. In: Hetnarski, R.B. (ed), Encyclopedia of Thermal Stresses, 11 vols., 1st edn., pp. 2677–2687. Springer, Berlin (2014)Google Scholar
  176. 341.
    Svanadze, M.: Potentials in thermoelasticity theory. In: Hetnarski, R.B. (ed.), Encyclopedia of Thermal Stresses, 11 vols., 1st edn., pp. 4013–4023. Springer, Berlin (2014)Google Scholar
  177. 342.
    Svanadze, M.: External boundary value problems of steady vibrations in the theory of rigid bodies with a double porosity structure. Proc. Appl. Math. Mech. 15(1), 365–366 (2015)MathSciNetCrossRefGoogle Scholar
  178. 343.
    Svanadze, M.: Plane waves, uniqueness theorems and existence of eigenfrequencies in the theory of rigid bodies with a double porosity structure. In: Albers, B., Kuczma, M. (eds.) Continuous Media with Microstructure, vol. 2, pp. 287–306. Springer, Basel (2016)CrossRefGoogle Scholar
  179. 344.
    Svanadze, M.: On the linear theory of thermoelasticity for triple porosity materials. In: Ciarletta, M., Tibullo, V., Passarella, F. (eds), Proceedings of 11th International Congress Thermal Stresses, 5–9 June, 2016, Salerno, Italy, pp. 259–262 (2016)Google Scholar
  180. 345.
    Svanadze, M.: Fundamental solutions in the theory of elasticity for triple porosity materials. Meccanica 51, 1825–1837 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  181. 346.
    Svanadze, M.: Boundary value problems in the theory of thermoelasticity for triple porosity materials. In: Proceedings of ASME2016. 50633; Vol. 9: Mechanics of Solids, Structures and Fluids; NDE, Diagnosis, and Prognosis, V009T12A079. November 11, 2016, IMECE2016-65046 (2016).  https://doi.org/10.1115/IMECE2016-65046
  182. 347.
    Svanadze, M.: Boundary value problems of steady vibrations in the theory of thermoelasticity for materials with double porosity structure. Arch. Mech. 69, 347–370 (2017)MathSciNetzbMATHGoogle Scholar
  183. 348.
    Svanadze, M.: External boundary value problems in the quasi static theory of thermoelasticity for triple porosity materials. Proc. Appl. Math. Mech. 17(1), 471–472 (2017)CrossRefGoogle Scholar
  184. 349.
    Svanadze, M.: Steady vibrations problems in the theory of elasticity for materials with double voids. Acta Mech. 229, 1517–1536 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  185. 350.
    Svanadze, M.: Potential method in the theory of elasticity for triple porosity materials. J. Elast. 130, 1–24 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  186. 351.
    Svanadze, M.: Potential method in the linear theory of triple porosity thermoelasticity. J. Math. Anal. Appl. 461, 1585–1605 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  187. 352.
    Svanadze, M.: On the linear equilibrium theory of elasticity for materials with triple voids. Quart. J. Mech. Appl. Math. 71, 329–248 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  188. 353.
    Svanadze, M.: External boundary value problems in the quasi static theory of thermoelasticity for materials with triple voids. Proc. Appl. Math. Mech. 18(1), e201800171 (2018)CrossRefGoogle Scholar
  189. 354.
    Svanadze, M.: Fundamental solutions in the linear theory of thermoelasticity for solids with triple porosity. Math. Mech. Solids 24, 919–938 (2019)MathSciNetCrossRefGoogle Scholar
  190. 355.
    Svanadze, M.: On the linear theory of double porosity thermoelasticity under local thermal non-equilibrium. J. Therm. Stresses 42, 890–913 (2019)CrossRefGoogle Scholar
  191. 356.
    Svanadze, M.: Potential method in the theory of thermoelasticity for materials with triple voids. Arch. Mech. 71, 113–136 (2019)MathSciNetGoogle Scholar
  192. 357.
    Svanadze, M., de Boer, R.: On the representations of solutions in the theory of fluid-saturated porous media. Quart. J. Mech. Appl. Math. 58, 551–562 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  193. 358.
    Svanadze, M., De Cicco, S.: Fundamental solutions in the full coupled linear theory of elasticity for solid with double porosity. Arch. Mech. 65, 367–390 (2013)MathSciNetGoogle Scholar
  194. 359.
    Svanadze, M., Scalia, A.: Mathematical problems in the coupled linear theory of bone poroelasticity. Comput. Math. Appl. 66, 1554–1566 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  195. 360.
    Svanadze, M., Scalia, A.: Potential method in the theory of thermoelasticity with microtemperatures for microstretch solids. Trans. Nanjing Univ. Aeron. Astron. 31, 159–163 (2014)Google Scholar
  196. 366.
    Svanadze, M.M.: External boundary value problems in the quasi static theory of viscoelasticity for Kelvin-Voigt materials with double porosity. Proc. Appl. Math. Mech. 16(1), 497–498 (2016)CrossRefGoogle Scholar
  197. 370.
    Svanadze, M.M.: Fundamental solutions and uniqueness theorems in the theory of viscoelasticity for materials with double porosity. Trans. A. Razmadze Math. Inst. 172, 276–292 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  198. 375.
    Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics. Handbuch der Physik, Band III/3, Flügge, S. (ed.). Springer, Berlin (1965)Google Scholar
  199. 376.
    Truesdell, C., Toupin, R.: The Classical Field Theories. Handbuch der Physik, Band III/1, Flügge, S. (ed.). Springer, Berlin (1960)Google Scholar
  200. 377.
    Tsagareli, I.: Explicit solution of elastostatic boundary value problems for the elastic circle with voids. Adv. Math. Phys. 2018, 6275432, 6pp. (2018). https://doi.org/10.1155/2018/6275432
  201. 378.
    Tsagareli, I., Bitsadze, L.: Explicit solution of one boundary value problem in the full coupled theory of elasticity for solids with double porosity. Acta Mech. 26, 1409–1418 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  202. 379.
    Tsagareli, I., Bitsadze, L.: Explicit solutions on some problems in the fully coupled theory of elasticity for a circle with double porosity. Bull. TICMI 20, 11–23 (2016)MathSciNetzbMATHGoogle Scholar
  203. 381.
    Tsagareli, I., Svanadze, M.M.: Explicit solution of the problems of elastostatics for an elastic circle with double porosity. Mech. Res. Commun. 46, 76–80 (2012)CrossRefGoogle Scholar
  204. 383.
    Vafai, K.: Porous Media: Applications in Biological Systems and Biotechnology. CRC Press, Boca Raton (2011)Google Scholar
  205. 389.
    Verruijt, A.: Theory and Problems of Poroelasticity. Delft University of Technology, Delft (2015)Google Scholar
  206. 392.
    Wang, H.F.: Theory of Linear Poro-Elasticity with Applications to Geomechanics and Hydrogeology. Princeton University Press, Princeton (2000)Google Scholar
  207. 394.
    Warren, J.R., Root, P.J.: The behaviour of naturally fractured reservoirs. Soc. Pet. Eng. J. 228, 245–255 (1963)CrossRefGoogle Scholar
  208. 395.
    Wei, Z., Zhang, D.: Coupled fluid - flow and geomechanics for triple - porosity/dual - permeability modelling of coalbed methane recovery. Int. J. Rock Mech. Min. Sci. 47, 1242–1253 (2008)CrossRefGoogle Scholar
  209. 396.
    Wilson, R.K., Aifantis, E.C.: On the theory of consolidation with double porosity - I. Int. J. Eng. Sci. 20, 1009–1035 (1982)zbMATHCrossRefGoogle Scholar
  210. 397.
    Wu, Y.-S.: Multiphase Fluid Flow in Porous and Fractured Reservoirs. Elsevier, Amsterdam (2016)Google Scholar
  211. 398.
    Wu, Y.-S., Liu, H.H., Bodavarsson, G.S.: A triple-continuum approach for modelling flow and transport processes in fractured rock. J. Contam. Hydrol. 73, 145–179 (2004)CrossRefGoogle Scholar
  212. 403.
    Zhang, W., Xu, J., Jiang, R., Cui, Y., Qiao, J., Kang, C., Lu, Q.: Employing a quad-porosity numerical model to analyze the productivity of shale gas reservoir. J. Petrol. Sci. Eng. 157, 1046–1055 (2017)CrossRefGoogle Scholar
  213. 404.
    Zhao, Y., Chen, M.: Fully coupled dual-porosity model for anisotropic formations. Int. J. Rock Mech. Min. Sci. 43, 1128–1133 (2006)CrossRefGoogle Scholar
  214. 405.
    Zou, M., Wei, C., Yu, H., Song, L.: Modelling and application of coalbed methane recovery performance based on a triple porosity/dual permeability model. J. Nat. Gas Sci. Eng. 22, 679–688 (2015)CrossRefGoogle Scholar
  215. 367.
    Svanadze, M.M.: Plane waves and problems of steady vibrations in the theory of viscoelasticity for Kelvin-Voigt materials with double porosity. Arch. Mech. 68, 441–458 (2016)MathSciNetzbMATHGoogle Scholar
  216. 368.
    Svanadze, M.M.: Fundamental solution and uniqueness theorems in the linear theory of thermoviscoelasticity for solids with double porosity. J. Therm. Stresses 40, 1339–1352 (2017)CrossRefGoogle Scholar
  217. 369.
    Svanadze, M.M.: External boundary value problems in the quasi static theory of thermoviscoelasticity for Kelvin-Voigt materials with double porosity. Proc. Appl. Math. Mech. 17(1), 469–470 (2017)CrossRefGoogle Scholar
  218. 96.
    Ciarletta, M., Straughan, B., Zampoli, V.: Thermo-poroacoustic acceleration waves in elastic materials with voids without energy dissipation. Int. J. Eng. Sci. 45, 736–743 (2007)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Merab Svanadze
    • 1
  1. 1.Ilia State UniversityTbilisiGeorgia

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