Fundamental Solutions for a Poroelastic Half-Space With Compressible Constituents

Abstract

This paper presents a comprehensive analytical treatment of the three-dimensional response of a poroelastic half-space with compressible constituents. General solutions for equations of equilibrium expressed in terms of displacements and variation of fluid volume are derived by applying Fourier expansion, Hankel transforms, and Laplace transforms with respect to the circumferential, radial, and time coordinates, respectively. The general solutions are used to derive a set of fundamental solutions corresponding to circular ring loads and to a fluid source applied at a finite depth below the free surface of a poroelastic half-space. The circumferential variation of the ring loads and the fluid source is described by appropriate trigonometric terms. Fundamental solutions presented in this paper can be used to model arbitrarily distributed loadings and a fluid source as well as a number of other problems encountered in geomechanics and energy resource explorations. In addition, these fundamental solutions can be used as the kernel functions in the development of boundary integral equation methods for a poroelastic half-space. Solutions for buried circular patch loads/fluid sink and concentrated loads/fluid sink are also deduced. Numerical evaluation of the fundamental solutions is also discussed. Comparisons are presented for time domain solutions obtained by using the Stehfest’s and Schapery’s schemes for Laplace inversion. A selected set of numerical solutions are presented to portray the features of consolidation process in six different poroelastic materials under buried patch loads and a sink.