Effect of Hydrological Properties on the Energy Shares of Reflected&nbsp;Waves at the Surface of a Partially Saturated Porous Solid

Mahabir Barak; Manjeet Kumari; Manjeet Kumar; Mahabir Barak; Manjeet Kumari; Manjeet Kumar

doi:10.3934/geosci.2017.1.67

AIMS Geosciences

2017, Volume 3, Issue 1: 67-90. doi: 10.3934/geosci.2017.1.67

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Research article

Effect of Hydrological Properties on the Energy Shares of Reflected Waves at the Surface of a Partially Saturated Porous Solid

1.
Dept. of Mathematics, Indira Gandhi University Meerpur, India-122503
2.
Dept. of Mathematics, C.M.R.J Govt. College, Ellenabad, India-125102

Received: 22 October 2016 Accepted: 23 January 2017 Published: 25 February 2017

In the present study, the reflection of inhomogeneous waves is investigated at the stress-free plane surface based on multiphase poroelasticity theory. The porous medium is considered as dissipative due to the presence of viscosity in pores fluid. Four inhomogeneous (i.e. different direction of propagation and attenuation) reflected waves (three longitudinal and one shear) exists due to an incident wave. By using the appropriate boundary conditions, closed-form analytical expressions for the reflection coeffcients are derived at the stress-free surface. These reflection coeffcients are used to drive the analytical expressions for the energy shares of various reflected inhomogeneous waves. In mathematical framework, the conservation of incident energy is confirmed by considering an interaction energy between two dissimilar waves. It validates that the numerical calculations are analytically correct. Finally, a numerical example is considered to study the effects of viscous cross-coupling, porosity, saturation of gas, pore-characteristics and wave frequency on the energy shares of various reflected inhomogeneous waves and depicted graphically.
- viscous cross-coupling,
- partially saturated porous solid,
- immiscible pore-fluids,
- inhomogeneous wave,
- reflection coefficients
Citation: Mahabir Barak, Manjeet Kumari, Manjeet Kumar. Effect of Hydrological Properties on the Energy Shares of Reflected Waves at the Surface of a Partially Saturated Porous Solid[J]. AIMS Geosciences, 2017, 3(1): 67-90. doi: 10.3934/geosci.2017.1.67

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Abstract

In the present study, the reflection of inhomogeneous waves is investigated at the stress-free plane surface based on multiphase poroelasticity theory. The porous medium is considered as dissipative due to the presence of viscosity in pores fluid. Four inhomogeneous (i.e. different direction of propagation and attenuation) reflected waves (three longitudinal and one shear) exists due to an incident wave. By using the appropriate boundary conditions, closed-form analytical expressions for the reflection coeffcients are derived at the stress-free surface. These reflection coeffcients are used to drive the analytical expressions for the energy shares of various reflected inhomogeneous waves. In mathematical framework, the conservation of incident energy is confirmed by considering an interaction energy between two dissimilar waves. It validates that the numerical calculations are analytically correct. Finally, a numerical example is considered to study the effects of viscous cross-coupling, porosity, saturation of gas, pore-characteristics and wave frequency on the energy shares of various reflected inhomogeneous waves and depicted graphically.

References

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