Analytical solution for the dynamic response of a saturated poroelastic half-space to harmonic stress loading

Abstract

A theoretical analysis for the dynamic response of a semi-infinite fluid-bearing porous medium to external harmonic loading is presented in this study based on the decoupled poroelasticity equations of Biot (1962). A corresponding initial and boundary value problem is formulated and the analytical solution for the induced pore pressure and total dilatational stress is determined using the technique of Laplace transforms. To investigate the quantitative impact of inertial effect on the poroelastic response, the problem is also solved analytically in the diffusive model (i.e. inertial terms are ignored). Comparison of the analytical solutions obtained from two different models shows that as inertial effect is accounted for, the response undergoes in a dynamic manner but lags behind the loading by a physical factor equal to the travel time necessary for pressure wave to reach the location that is prescribed. A numerical study is then conducted for water-containing Columbia fine sandy loam at lower excitation frequencies as a representative example. Our numerical results reveal that the dynamic model yields a cyclic response for the induced pore pressure and total dilatational stress with respect to depth, but the diffusive model fails to predict this attribute. Lastly, we find in the dynamic model that effective stress may take on a positive value at some depths due to the existence of time lag in the response of pore fluid to external loading so that the solid skeleton needs to sustain excess fluid pressure. This positive value is crucial for the phenomenon of liquefaction if the loading is substantial enough.