A semi-analytical solution for the one-dimensional transient response of layered unsaturated porous media

The transient response of porous media is an important aspect of dynamic research. However, existing studies seldom provide solutions to the transient response problem of layered unsaturated porous media. Based on the Biot-type unsaturated wave equations, dimensionless one-dimensional wave equations are established. An appropriate displacement function is introduced to homogenize the boundary conditions. Subsequently, the transfer matrix method is used to obtain the eigenvalues and eigenfunctions of the homogeneous governing equations. Leveraging the orthogonality of the eigenfunctions, the original problem is transformed into solving a series of initial value problems of ordinary differential equations. The temporal solution within the time domain is then obtained through an improved precise time integration method. The validity of the solution presented in this paper is verified by comparing it with existing solutions in the literature. Analysis of numerical examples shows that reflection waves of opposite phases will be generated at the hard-soft and hard-harder interface, which helps in the accurate identification of weak interlayers in practical engineering applications. With increasing saturation, there is a noticeable increase in the velocities of the P1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{1}$$\end{document} and P3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{3}$$\end{document} waves, whereas the velocity of the P2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{2}$$\end{document} waves tends to decrease, which can be used to assess the mechanical property of medium. The peak value of pore pressure in unsaturated can be 1.64 times higher than those in saturated condition.