A Semianalytical Solution for the Transient Response of One-Dimensional Unsaturated Single-Layer Porous Media with General Boundary Conditions

Abstract

According to the governing wave equations for unsaturated porous media, a theoretical model is proposed to obtain the transient response solution of single-layer unsaturated porous media in one-dimensional conditions with general boundary conditions and arbitrary initial conditions when subjected to arbitrary vertical load. Parameters total stress σ, absolute soil displacement u, wetting fluid pressure pw, relative displacement w, nonwetting fluid pressure pn, and relative displacement v, which are independent of each other, are suggested firstly to express the boundary conditions through a linear combination method in pairs. Various boundary conditions can be obtained through adjusting the linear combination parameters. By utilizing the variable separation method, the eigenvalues and eigenfunctions of the undamped characteristic equation are obtained. Through the method of undetermined coefficients and orthogonality of eigenfunctions, the transient response problem can be converted to an initial value problem, which includes a series of ordinary differential equations. By making use of the precise time-integration method modified by the authors, the semianalytical solutions can finally be approached. Compared with existing research, the semianalytical solutions of this study are more general and can be exactly degenerated into some other boundary conditions. The validity of this method is verified by comparing the results with the current literature. Then, several examples are carried out to study the transient responses characteristic of unsaturated porous media subjected to the step load. The results demonstrate that there are three kinds of compression wave in the dynamic response of unsaturated porous media. In a short time, the semipermeable condition results are similar to permeable and impermeable conditions. However, results become complex over a long time due to the reflection and superposition of the compression waves.