Semi-analytical solutions for transient response of one-dimensional saturated single-layer porous media under various permeable boundary conditions

Based on the one-dimensional basic equations for waves in saturated porous media proposed by Biot, a mathematical model for one-dimensional transient response of single-layer saturated porous media is established under various permeable boundary conditions, arbitrary initial conditions and arbitrary vertical loads. Firstly, the independent parameters fluid pressure p and fluid relative displacement w are proposed to describe the various permeable boundary conditions by linear combination through adjusting the parameters. By utilizing the variable separation method, the eigen-values and the eigen-functions are obtained for undamped governing equations. With the help of undetermined coefficients and orthogonality of eigenfunctions methods, the solution to the problem can be converted to solve the initial value problem of a series of ordinary differential equations. The semi-analytical solutions are approached by using the precise time-integration method. Compared with those of the previous researches, the semi-analytical solutions of this research are more general and can be degenerated into various conditions exactly. Several numerical simulations are carried out to validate the proposed method. Finally, the one-dimensional transient responses of single-layer saturated soil with general boundary conditions under step loads are analyzed. The results demonstrate that the responses of semi-permeable condition are between the permeable and impermeable conditions. The displacements of solid and fluid increase first and then decrease. The incident and reflected waves stimulate the same phase pore pressure.