Surface waves in a general anisotropic poroelastic solid half-space

Abstract

SUMMARY A method is introduced for studying surface waves in a general anisotropic poroelastic medium. The method is analogous to the one used for isotropic media and derives a complex secular equation to represent the propagation of surface waves at the stress-free plane surface of a nondissipative porous medium. The point of importance is that the derived equation is, analytically, separable into real and imaginary parts and hence can be solved by iterative numerical methods. A root of this secular equation represents the existence of surface waves and calculates the apparent phase velocity along a given direction on the surface. Numerical work is carried out for the model of a crustal rock. The propagation of surface waves is studied numerically for the top three anisotropies (i.e. triclinic, monoclinic, orthorhombic). 1I N T R O DUCTION The dynamic behaviour of fluid-saturated porous media is attracting wide attention due to its importance in oil exploration, earthquake engineering, structural engineering, soil dynamics and hydrology. The dynamic equations formulated by Biot (1956) have long been regarded as standard and have formed the basis for the study of wave propagation problems in poroelastic media. Biot (1955) discussed the stress‐strain relations for anisotropic poroelasticity. Anisotropy in porous solids may be the result of propagation through a distribution of aligned cracks, microcracks and preferentially oriented pore space. Occasionally, anisotropy may be due some other phenomenon such as rock foliation or crystal alignment. Crampin (1994) reviewed the various studies related to the observation of shear wave splitting and confirmed the presence of anisotropy in almost all the rocks in the uppermost half of the crust. Thomsen (1995) related the anisotropy to pore properties and crack parameters in a porous rock. This work was supported by an experimental study of the anisotropy of sandstone with controlled crack geometry by Rathore et al. (1995). Sharma (1996) discussed the coexistence of cracks and pores and its effect on surface wave propagation. Hudson et al. (1996) studied the effects of connection between cracks and of small-scale porosity within the solid material on the overall elastic properties of cracked solids. The presence of mineral orientations, microfracturing, thin layering or combinations of these in a material results in a general anisotropy of arbitrary symmetry. The absence of symmetry in aligned microcracks or pore space also results in general anisotropy. The propagation of energy in such a general anisotropic medium is, in fact, a 3-D phenomenon. Wave motion in such a material is not confined to a particular plane. This explains the out-of-sagittal-plane motion of particles in Rayleigh waves. Apart from its well-known seismic importance, surface wave propagation in anisotropic media has widespread applications in the non-destructive evaluation of materials (Buden & Datta 1990; W uW Ting 2002) to study the anisotropic propagation of surface waves considers motion in an arbitrary but fixed plane. The Christoffel equation, in a polynomial form, is solved for vertical slowness values representing different quasiwaves. In the case of a general anisotropic porous solid, an eighth-order polynomial equation is required to be solved for complex values of vertical slowness. This gives a complex secular equation to represent surface waves in the medium. It is not surprising that there is no analytical method available to solve this system. Even the numerical solution may be very difficult. Moreover, researchers in this field may expect a method that is analogous to that used to study Rayleigh waves in an isotropic medium. This means that the secular equation should involve the elastic constants of the medium and the velocities (instead of the vertical slownesses) of the various body waves propagating in the medium. In this paper such a secular equation is derived for the surface waves in an anisotropic porous solid with arbitrary symmetry.