Scattering of monochromatic longitudinal waves on a planar crack of arbitrary shape in a fluid-saturated poroelastic medium

Abstract

Scattering of monochromatic longitudinal waves on a planar crack of arbitrary shape in a saturated poroelastic medium is considered. The medium is described by Biot’s constitutive equations, the crack sides are fluid permeable. The problem is reduced to a two-dimensional integral equation for the crack opening vector. Gaussian approximating functions are used for discretization of this equation. For such functions, the elements of the matrix of discretized problem are combinations of four standard one-dimensional integrals that can be tabulated. As a result, numerical integration is not needed. For regular grids of approximating nodes, this matrix has Toeplitz’s structure, and matrix-vector products can be calculated by the fast Fourier transform technique. The latter accelerates substantially the process of iterative solution of the discretized problem. Calculation of crack opening vectors, differential, and total cross-sections of circular and elliptic cracks are performed for longitudinal incident waves orthogonal to the crack surfaces. Dependencies of these characteristics on the medium permeability and wavefrequency are studied. Comparison of a crack in the poroelastic medium and in a dry elastic medium with the same porosity and skeleton elastic properties is presented.