Reflection and refraction of plane waves at the boundary of an elastic solid and double-porosity dual-permeability materials
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摘要
Phenomena of reflection and refraction of plane harmonic waves at a plane interface between an elastic solid and doubleporosity dual-permeability material are investigated. The elastic solid behaves non-dissipatively, while double-porosity dual-permeability materials behave dissipatively to wave propagation due to the presence of viscosity in pore fluids. All the waves(i.e., incident and reflected) in an elastic medium are considered as homogeneous(i.e., having the same directions of propagation and attenuation), while all the refracted waves in double-porosity dual-permeability materials are inhomogeneous(i.e., having different directions of propagation and attenuation). The coefficients of reflection and refraction for a given incident wave are obtained as a non-singular system of linear equations. The energy shares of reflected and refracted waves are obtained in the form of an energy matrix. A numerical example is considered to calculate the partition of incident energy among various reflected and refracted waves. The effect of incident direction on the partition of the incident energy is analyzed with a change in wave frequency, wave-induced fluid-flow, pore-fluid viscosity and double-porosity structure.It has been confirmed from numerical interpretation that during the reflection/refraction process, conservation of incident energy is obtained at each angle of incidence.
Phenomena of reflection and refraction of plane harmonic waves at a plane interface between an elastic solid and doubleporosity dual-permeability material are investigated. The elastic solid behaves non-dissipatively, while double-porosity dual-permeability materials behave dissipatively to wave propagation due to the presence of viscosity in pore fluids. All the waves(i.e., incident and reflected) in an elastic medium are considered as homogeneous(i.e., having the same directions of propagation and attenuation), while all the refracted waves in double-porosity dual-permeability materials are inhomogeneous(i.e., having different directions of propagation and attenuation). The coefficients of reflection and refraction for a given incident wave are obtained as a non-singular system of linear equations. The energy shares of reflected and refracted waves are obtained in the form of an energy matrix. A numerical example is considered to calculate the partition of incident energy among various reflected and refracted waves. The effect of incident direction on the partition of the incident energy is analyzed with a change in wave frequency, wave-induced fluid-flow, pore-fluid viscosity and double-porosity structure.It has been confirmed from numerical interpretation that during the reflection/refraction process, conservation of incident energy is obtained at each angle of incidence.
Phenomena of reflection and refraction of plane harmonic waves at a plane interface between an elastic solid and doubleporosity dual-permeability material are investigated. The elastic solid behaves non-dissipatively, while double-porosity dual-permeability materials behave dissipatively to wave propagation due to the presence of viscosity in pore fluids. All the waves(i.e., incident and reflected) in an elastic medium are considered as homogeneous(i.e., having the same directions of propagation and attenuation), while all the refracted waves in double-porosity dual-permeability materials are inhomogeneous(i.e., having different directions of propagation and attenuation). The coefficients of reflection and refraction for a given incident wave are obtained as a non-singular system of linear equations. The energy shares of reflected and refracted waves are obtained in the form of an energy matrix. A numerical example is considered to calculate the partition of incident energy among various reflected and refracted waves. The effect of incident direction on the partition of the incident energy is analyzed with a change in wave frequency, wave-induced fluid-flow, pore-fluid viscosity and double-porosity structure.It has been confirmed from numerical interpretation that during the reflection/refraction process, conservation of incident energy is obtained at each angle of incidence.
引文
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Ainslie MA, Burns PW. Energy-conserving reflection and transmission coefficients for a solid-solid boundary. J Acoust Soc Am.1995;98:2836-40. https://doi.org/10.1121/1.413249.
Ba J, Carcione JM, Nie JX. Biot-Rayleigh theory of wave propagation in double-porosity media. J Geophys Res. 2011;116:B06202. https://doi.org/10.1029/2010JB008185.
Bai R, Tinel A, Alem A, Franklin H, Wang H. Ultrasonic characterization of water saturated double porosity media. Phys Procedia.2015;70:114-7. https://doi.org/10.1016/j.phpro.2015.08.055.
Bai R, Tinel A, Alem A, Franklin H, Wang H. Estimating frame bulk and shear moduli of two double porosity layers by ultrasound transmission. Ultrasonics. 2016;70:211-20. https://doi.org/10.1016/j.ultras.2016.05.004.
Batzle ML, Han DH, Hofmann R. Fluid mobility and frequencydependent seismic velocity-direct measurements. Geophysics.2006;71(1):1-9. https://doi.org/10.1190/1.2159053.
Berryman JG, Wang HF. The elastic coefficients of double-porosity models for fluid transport in jointed rock. J Geophys Res.1995;100:34611-27. https://doi.org/10.1029/95JB02161.
Berryman JG, Wang HF. Elastic wave propagation and attenuation in a double-porosity dual-permeability medium. Int J Rock Mech Min Sci. 2000;37:63-78. https://doi.org/10.1016/S1365-1609(99)00092-1.
Bhagwan J, Tomar SK. Reflection and transmission of plane dilatational wave at a plane interface between an elastic solid halfspace and a thermo-viscoelastic solid half-space with voids.J Elast. 2015;121(1):69-88. https://doi.org/10.1007/s10659-015-9522-9.
Borcherdt RD. Viscoelastic waves in layered media. New York:Cambridge University Press; 2009. https://doi.org/10.1121/1.3243311.
Bullen KE. An introduction to the theory of seismology. England:Cambridge University Press; 1962.
Dai ZJ, Kuang ZB, Zhao SX. Reflection and transmission of elastic waves at the interface between an elastic solid and a double porosity medium. Int J Rock Mech Min Sci. 2006a;43:961-71. https://doi.org/10.1016/j.ijrmms.2005.11.010.
Dai ZJ, Kuang ZB, Zhao SX. Reflection and transmission of elastic waves from the interface of fluid saturated porous solid and a double porosity solid. Transp Porous Med. 2006b;65:237-64. https://doi. org/10.1007/s 11242-007-9155-y.
Dai ZJ, Kuang ZB. Reflection and transmission of elastic waves at the interface between water and a double porosity solid. Transp Porous Med. 2008;72(3):369-92. https://doi.org/10.1007/s11242-005-6084-5.
Deresiewicz H, Skalak R. On uniqueness in dynamic poroelasticity.Bull Seismol Soc Am. 1963;53(4):783-8.
Goyal S, Tomar SK. Reflection/refraction of a dilatational wave at a plane interface between uniform elastic and swelling porous half-spaces. Transp Porous Med. 2015;109(3):609-32. https://doi.org/10.1007/s 11242-015-0539-0.
Krebes ES. The viscoelastic reflection/transmission problem:two special cases. Bull Seismol Soc Am. 1983;73:1673-83.
Kumar M, Saini R. Reflection and refraction of attenuated waves at the boundary of elastic solid and porous solid saturated with two immiscible viscous fluids. Appl Math Mech Engl Ed.2012;33(6):797-816. https://doi.org/10.1007/s10483-012-1587-6.
Kumar M, Saini R. Reflection and refraction of waves at the boundary of non viscous porous solid saturated with single fluid and a porous solid saturated with immiscible fluids. Lat Am J Solids Struct. 2016;13:1299-324. https://doi.org/10.1590/1679-78252090.
Kumar M, Sharma MD. Reflection and transmission of attenuated waves at the boundary between two dissimilar poroelastic solids saturated with two immiscible fluids. Geophys Prospect.2013;61(5):1035-55. https://doi.org/10.1111/1365-2478.12049.
Pride SR, Berryman JG. Linear dynamics of double porosity dual-permeability materials:I. Governing equations and acoustic attenuation. Phys Rev E. 2003a;68(3):036603. https://doi.org/10.1103/Phy sRevE.68.036603.
Pride SR, Berryman JG. Linear dynamics of double porosity dualpermeability materials:Ⅱ. Fluid transport equations. Phys Rev E.2003b;68:036604. https://doi.org/10.1103/PhysRevE.68.036604.
Pride SR, Berryman JG, Harris JM. Seismic attenuation due to waveinduced flow. J Geophys Res. 2004;109:B01201. https://doi.org/10.1029/2003JB002639.
Sharma MD. Boundary conditions for porous solids saturated with viscous fluid. Appl Math Mech Engl Ed. 2009;30(7):821-32. https://doi.org/10.1007/s10483-009-0702-6.
Sharma MD. Effect of local fluid flow on reflection of plane elastic waves at the boundary of a double-porosity medium. Adv Water Resour. 2013;61:62-73. https://doi.org/10.1016/j.advwa tres.2013.09.001.
Sharma MD. Effect of local fluid flow on Rayleigh waves in a double porosity solid. Bull Seismol Soc Am. 2014;104(6):2633-43. https://doi. org/10.1785/0120140014.
Sharma MD. Effect of local fluid flow on the propagation of elastic waves in a transversely isotropic double-porosity medium. Geophys J Int. 2015a;200:1423-35. https://doi.org/10.1093/gji/ggu485.
Sharma MD. Constitutive relations for wave propagation in a double porosity solid. Mech Mater. 2015b;91:263-76. https://doi.org/10.1016/j.mechmat.2015.08.005.
Sharma MD. Wave-induced flow of pore fluid in a double-porosity solid under liquid layer. Transp Porous Med. 2016;113(3):531-47.https://doi.org/10.1007/s11242-016-0709-8.
Sharma MD. Wave propagation in double-porosity dual porosity materials:velocity and attenuation. Adv Water Resour. 2017a;106:132-43. https://doi.org/10.1016/j.advwatres.2017.02.016.
Sharma MD. Propagation and attenuation of inhomogeneous waves in double-porosity dual-permeability materials. Geophys J Int.2017b;208(2):737-47. https://doi.org/10.1093/gji/ggw423.
Shekhar S, Parvez IA. Wave propagation across the imperfectly bonded interface between cracked elastic solid and porous solid saturated with two immiscible viscous fluids. Int J Solids Struct. 2015;75(75-76):299-308. https://doi.org/10.1016/j.ijsol str.2015.08.022.
Shekhar S, Parvez IA. Reflection and refraction of attenuated waves at the interface between cracked poroelastic medium and porous solid saturated with two immiscible fluids. Transp Porous Med.2016;113(2):405-30. https://doi.org/10.1007/s11242-016-0704-0.
Sun W, Ba J, Carcione JM. Theory of wave propagation in partially saturated double-porosity rocks:a triple-layer patchy model. Geophys J Int. 2016;205:22-37. https://doi.org/10.1093/gji/ggv551.
Tomar SK, Arora A. Reflection and transmission of elastic waves at an elastic/porous solid saturated by two immiscible fluids. Int J Solids Struct. 2006;43:1991-2013. https://doi.org/10.1016/j.ijsol str.2005.05.056.
Yeh CL,Lo WC,Jan CD,Yang CC. Reflection and refraction of obliquely incident elastic waves upon the interface between two porous elastic half-spaces saturated by different fluid mixtures.J Hydrol. 2010;395:91-102. https://doi.org/10.1016/j.jhydr ol.2010.10.018.
Zheng P,Ding B,Sun X. Elastic wave attenuation and dispersion induced by mesoscopic flow in double-porosity rocks. Int J Rock Mech Min Sci. 2017;91:104-11. https://doi.org/10.1016/j.ijrmm s.2016.11.018.
Ainslie MA, Burns PW. Energy-conserving reflection and transmission coefficients for a solid-solid boundary. J Acoust Soc Am.1995;98:2836-40. https://doi.org/10.1121/1.413249.
Ba J, Carcione JM, Nie JX. Biot-Rayleigh theory of wave propagation in double-porosity media. J Geophys Res. 2011;116:B06202. https://doi.org/10.1029/2010JB008185.
Bai R, Tinel A, Alem A, Franklin H, Wang H. Ultrasonic characterization of water saturated double porosity media. Phys Procedia.2015;70:114-7. https://doi.org/10.1016/j.phpro.2015.08.055.
Bai R, Tinel A, Alem A, Franklin H, Wang H. Estimating frame bulk and shear moduli of two double porosity layers by ultrasound transmission. Ultrasonics. 2016;70:211-20. https://doi.org/10.1016/j.ultras.2016.05.004.
Batzle ML, Han DH, Hofmann R. Fluid mobility and frequencydependent seismic velocity-direct measurements. Geophysics.2006;71(1):1-9. https://doi.org/10.1190/1.2159053.
Berryman JG, Wang HF. The elastic coefficients of double-porosity models for fluid transport in jointed rock. J Geophys Res.1995;100:34611-27. https://doi.org/10.1029/95JB02161.
Berryman JG, Wang HF. Elastic wave propagation and attenuation in a double-porosity dual-permeability medium. Int J Rock Mech Min Sci. 2000;37:63-78. https://doi.org/10.1016/S1365-1609(99)00092-1.
Bhagwan J, Tomar SK. Reflection and transmission of plane dilatational wave at a plane interface between an elastic solid halfspace and a thermo-viscoelastic solid half-space with voids.J Elast. 2015;121(1):69-88. https://doi.org/10.1007/s10659-015-9522-9.
Borcherdt RD. Viscoelastic waves in layered media. New York:Cambridge University Press; 2009. https://doi.org/10.1121/1.3243311.
Bullen KE. An introduction to the theory of seismology. England:Cambridge University Press; 1962.
Dai ZJ, Kuang ZB, Zhao SX. Reflection and transmission of elastic waves at the interface between an elastic solid and a double porosity medium. Int J Rock Mech Min Sci. 2006a;43:961-71. https://doi.org/10.1016/j.ijrmms.2005.11.010.
Dai ZJ, Kuang ZB, Zhao SX. Reflection and transmission of elastic waves from the interface of fluid saturated porous solid and a double porosity solid. Transp Porous Med. 2006b;65:237-64. https://doi. org/10.1007/s 11242-007-9155-y.
Dai ZJ, Kuang ZB. Reflection and transmission of elastic waves at the interface between water and a double porosity solid. Transp Porous Med. 2008;72(3):369-92. https://doi.org/10.1007/s11242-005-6084-5.
Deresiewicz H, Skalak R. On uniqueness in dynamic poroelasticity.Bull Seismol Soc Am. 1963;53(4):783-8.
Goyal S, Tomar SK. Reflection/refraction of a dilatational wave at a plane interface between uniform elastic and swelling porous half-spaces. Transp Porous Med. 2015;109(3):609-32. https://doi.org/10.1007/s 11242-015-0539-0.
Krebes ES. The viscoelastic reflection/transmission problem:two special cases. Bull Seismol Soc Am. 1983;73:1673-83.
Kumar M, Saini R. Reflection and refraction of attenuated waves at the boundary of elastic solid and porous solid saturated with two immiscible viscous fluids. Appl Math Mech Engl Ed.2012;33(6):797-816. https://doi.org/10.1007/s10483-012-1587-6.
Kumar M, Saini R. Reflection and refraction of waves at the boundary of non viscous porous solid saturated with single fluid and a porous solid saturated with immiscible fluids. Lat Am J Solids Struct. 2016;13:1299-324. https://doi.org/10.1590/1679-78252090.
Kumar M, Sharma MD. Reflection and transmission of attenuated waves at the boundary between two dissimilar poroelastic solids saturated with two immiscible fluids. Geophys Prospect.2013;61(5):1035-55. https://doi.org/10.1111/1365-2478.12049.
Pride SR, Berryman JG. Linear dynamics of double porosity dual-permeability materials:I. Governing equations and acoustic attenuation. Phys Rev E. 2003a;68(3):036603. https://doi.org/10.1103/Phy sRevE.68.036603.
Pride SR, Berryman JG. Linear dynamics of double porosity dualpermeability materials:Ⅱ. Fluid transport equations. Phys Rev E.2003b;68:036604. https://doi.org/10.1103/PhysRevE.68.036604.
Pride SR, Berryman JG, Harris JM. Seismic attenuation due to waveinduced flow. J Geophys Res. 2004;109:B01201. https://doi.org/10.1029/2003JB002639.
Sharma MD. Boundary conditions for porous solids saturated with viscous fluid. Appl Math Mech Engl Ed. 2009;30(7):821-32. https://doi.org/10.1007/s10483-009-0702-6.
Sharma MD. Effect of local fluid flow on reflection of plane elastic waves at the boundary of a double-porosity medium. Adv Water Resour. 2013;61:62-73. https://doi.org/10.1016/j.advwa tres.2013.09.001.
Sharma MD. Effect of local fluid flow on Rayleigh waves in a double porosity solid. Bull Seismol Soc Am. 2014;104(6):2633-43. https://doi. org/10.1785/0120140014.
Sharma MD. Effect of local fluid flow on the propagation of elastic waves in a transversely isotropic double-porosity medium. Geophys J Int. 2015a;200:1423-35. https://doi.org/10.1093/gji/ggu485.
Sharma MD. Constitutive relations for wave propagation in a double porosity solid. Mech Mater. 2015b;91:263-76. https://doi.org/10.1016/j.mechmat.2015.08.005.
Sharma MD. Wave-induced flow of pore fluid in a double-porosity solid under liquid layer. Transp Porous Med. 2016;113(3):531-47.https://doi.org/10.1007/s11242-016-0709-8.
Sharma MD. Wave propagation in double-porosity dual porosity materials:velocity and attenuation. Adv Water Resour. 2017a;106:132-43. https://doi.org/10.1016/j.advwatres.2017.02.016.
Sharma MD. Propagation and attenuation of inhomogeneous waves in double-porosity dual-permeability materials. Geophys J Int.2017b;208(2):737-47. https://doi.org/10.1093/gji/ggw423.
Shekhar S, Parvez IA. Wave propagation across the imperfectly bonded interface between cracked elastic solid and porous solid saturated with two immiscible viscous fluids. Int J Solids Struct. 2015;75(75-76):299-308. https://doi.org/10.1016/j.ijsol str.2015.08.022.
Shekhar S, Parvez IA. Reflection and refraction of attenuated waves at the interface between cracked poroelastic medium and porous solid saturated with two immiscible fluids. Transp Porous Med.2016;113(2):405-30. https://doi.org/10.1007/s11242-016-0704-0.
Sun W, Ba J, Carcione JM. Theory of wave propagation in partially saturated double-porosity rocks:a triple-layer patchy model. Geophys J Int. 2016;205:22-37. https://doi.org/10.1093/gji/ggv551.
Tomar SK, Arora A. Reflection and transmission of elastic waves at an elastic/porous solid saturated by two immiscible fluids. Int J Solids Struct. 2006;43:1991-2013. https://doi.org/10.1016/j.ijsol str.2005.05.056.
Yeh CL,Lo WC,Jan CD,Yang CC. Reflection and refraction of obliquely incident elastic waves upon the interface between two porous elastic half-spaces saturated by different fluid mixtures.J Hydrol. 2010;395:91-102. https://doi.org/10.1016/j.jhydr ol.2010.10.018.
Zheng P,Ding B,Sun X. Elastic wave attenuation and dispersion induced by mesoscopic flow in double-porosity rocks. Int J Rock Mech Min Sci. 2017;91:104-11. https://doi.org/10.1016/j.ijrmm s.2016.11.018.