Scattering of plane monochromatic waves from a heterogeneous inclusion of arbitrary shape in a poroelastic medium: An efficient numerical solution

Scattering of plane longitudinal monochromatic waves from a heterogeneous inclusion of arbitrary shape in an infinite poroelastic medium is considered. Wave propagation in the medium is described by Biot’s equations of poroelasticity. The scattering problem is formulated in terms of the volume integral equations for displacements of the solid skeleton and fluid pressure in the pore space in the region occupied by the inclusion. An efficient numerical method is applied to solve these equations. In the method, Gaussian approximating functions are used for discretization of the problem. For regular node grids, the matrix of the discretized problem has Toeplitz’s properties, and the Fast Fourier Transform technique can be used for the calculation of matrix–vector products. The latter accelerates substantially the process of iterative solution of the discretized problem. For material parameters of typical sedimentary rocks, the system of differential equations of poroelasticity contains a differential operator with a small parameter. As the result, the wave field in the inclusion region is split up into a slowly changing part, and boundary layer functions concentrated near the inclusion interface. The method of matched asymptotic expansions is used for the numerical solution in this case. For a spherical inclusion, the results of the numerical and matched asymptotic expansion methods are compared with a semi-analytical series solution. For a non-spherical heterogeneous inclusion, an example of the numerical solution is presented.