Lamb's problem for fluid-saturated, porous media

An analytic solution of the axisymmetric propagation in a fluid-saturated half-space produced by the concentrated load P oexp( iωt ) acting vertically at the surface is presented for the first time. We start with a derivation of the field equations in cylindrical coordinates on the basis of Biot's linear theory for porous, fluid-saturated media. The field equations are then reduced to Helmoholtz-type equations with the aid of four scalar potentials related to the solid and fluid constituents of the two-phased medium. Two uncoupled (dilatation-rotation) sets of equations are obtained, each of them consisting of two coupled (solid-fluid) equations. By solving the governing homogeneous wave equations, we obtain two complex eigenvalue problems which are associated with the propagation of free annular waves in a fluid-saturated, porous medium. Using the homogeneous solution, a boundary-value problem is formulated in the frequency-wavenumber ( f-k ) domain from which integral solutions for the surface displacements are obtained. It is shown that, in the absence of a saturating pore fluid, the results are reduced to known solutions of Lamb's problem for a conventional one-phased medium. Consistent with the two-phased nature of the problem, our solution considers two types of dissipation. The first is due to intergranular energy losses in the solid phase, while the second is due to viscous resistance to the flow of the pore fluid. Plots of the surface displacements are presented in the f-k domain which compare “saturating” with “dry” cases. It is shown that the effect of the pore fluid is to reduce the amplitude and frequency of the dominant mode of the response. These effects, however, are found to be pronounced only when dissipation due to the flow of the pore fluid is considered in the solution. If the latter type of dissipation is omitted, then the conventional solution of Lamb's problem for one-phased half-space is sufficient for practical purposes.