Exact Solutions For One-dimensional Problems Of Saturated And Unsaturated Porous Media

Research on the static and dynamic responses of porous media is an important issue in areas such as geotechnical, earthquake, oceanographic engineering, and resources exploration and development. Due to the complexity of the inertial, viscous and mechanical couplings, the difficulty of the research on the wave propagation in porous media is much higher than that in single phase media. Most of the problems of fluid-saturated and unsaturated porous media can often be predicted quantitatively via numerical methods involving discretisation of both spatial and temporal domains. Even in the one-dimensional problem, only a few analytical solutions are available in the literature. However, thorough understanding of the one-dimensional static and dynamic responses of porous media is the foundation of studing that of porous media; therefore, it is worthy to study the solution for one-dimensional transient response of fluid-saturated porous media.The exact solution can not only illustrate the static and dynamic responses of porous media, but also be used to evaluate the validity and accuracy of various numerical results. Using the method of transforming a non-homogeneous boundary condition into a homogeneous one, the separation of variable method or the eigenfunction expansion method, the Fourier Sine and Cosine transform method, and the state space method, the one-dimensional consolidation of unsaturated soil and the one-dimensional transient response of porous media are studied.For the one-dimensional consolidation of unsaturated single-layer soil, based on the Fredlund one-dimensional consolidation equation, in which the water permeability and the air transmission are assumed to be constants. Series solutions for ten kinds of boundary conditions are obtained. For the one-dimensional transient responses of saturated single-layer porous media and saturated semi-infinite porous media, based on the Biot equation of saturated porous media, series solutions for three kinds of boundary conditions and integral expressions of the solutions for two kinds of boundary conditions are obtained, respectively. For the one-dimensional transient responses of imcompressible saturated single-layer porous media and saturated semi-infinite porous media, based on the Biot equation of imcompressible saturated porous media, series solutions for four kinds of boundary conditions and integral expressions of the solutions for two kinds of boundary conditions are obtained, respectively. For the one-dimensional transient responses of unsaturated single-layer porous media and unsaturated semi-infinite porous media, based on the Zienkiewicz equation of unsaturated porous media, series solutions for three kinds of boundary conditions and integral expressions of the solutions for two kinds of boundary conditions are obtained, respectively.Finally, the exact solutions are compared with the solutions obtained by other methods to validate the correctness of exact solutions. Through numerical examples, the one-dimensional consolidation behaviour of unsaturated soil and the propagation behaviour of compressional waves in saturated porous media, incompressible saturated porous media, and unsaturated porous media are studied.