A semi-analytical solution for the transient response of one-dimensional saturated single-layer porous media with general boundary conditions

Based on the basic equations for saturated porous media wave equations proposed by Biot, a mathematical model for one-dimensional transient response of single-layer saturated porous media were established with general boundary condition, arbitrary initial conditions and arbitrary vertical load. Firstly, the independent parameters total stress σ, fluid pressure p, soil absolute displacement u and fluid relative displacement w were advised to describe the boundary conditions by linear combination in pairs. Through adjusting the parameters, the linear combination boundary conditions could represent various boundary conditions. The eigenvalues and the eigenfunctions of the undamped governing equations were obtained by means of the variable separation method. With the help of undetermined coefficients and orthogonality of eigenfunctions methods, the solution to the problem could be converted to solve the initial value problem of a series of ordinary differential equations. The semi-analytical solutions were approached by the precise time-integration method. Compared with previous research, the semi-analytical solutions of this research were more general and could be degenerated into various conditions exactly. Several numerical simulations were carried out to validate this method. Finally, the one-dimensional transient responses of single-layer saturated soil with general boundary conditions under step load were analyzed. The results demonstrate that the responses of semi-permeable condition are between permeable condition and impermeable condition and the solid and fluid displacement increase first and then decrease with time.