Reflection of attenuated waves at the surface of a porous solid saturated with two immiscible viscous fluids

Abstract

SUMMARY The mathematical model for wave motion in a porous solid saturated by two immiscible fluids is solved for the propagation of harmonic plane waves along a general direction in 3-D space. The solution is obtained in the form of Christoffel equations, which are solved further to calculate the complex velocities and polarisations of three longitudinal waves and one transverse wave. For any of these four attenuated waves, a general inhomogeneous propagation is considered through a particular specification of complex slowness vector. Inhomogeneity of an attenuated wave is represented through a finite non-dimensional parameter. For an arbitrarily chosen value of this inhomogeneity parameter, phase velocity and attenuation of a wave are calculated from the specification of its slowness vector. This specification enables to separate the contribution from homogeneous propagation of attenuated wave to the total attenuation. The phenomenon of reflection is studied to calculate the partition of wave-induced energy incident at the plane boundary of the porous solid. A parameter is used to define the partial opening of pores at the surface of porous solid. An arbitrary value of this parameter allows to study the variations in the energy partition with the opening of surface pores from fully closed to perfectly open. Another parameter is used to vary the saturation in pores from whole liquid to whole gas. Numerical examples are considered to discuss the effects of propagation direction, inhomogeneity parameter, opening of surface pores and saturating pore-fluid on the partition of incident energy.