Seepage problems in half-space domains are crucial in hydrology, environmental, and civil engineering, involving groundwater flow, pollutant transport, and structural stability. Typical examples include seepage through dam foundations, coastal aquifers, and levees under seepage forces, requiring accurate numerical modeling. However, existing methods face challenges in handling complex geometries, heterogeneous media, and anisotropic properties, particularly in multi-domain half-spaces. This study addresses these challenges by extending the modified scaled boundary finite element method (SBFEM) and using this method to explore steady seepage problems in complex half-space domain. In the modified SBFEM framework, segmented straight lines or curves, parallel to the far-field infinite boundary, are introduced as scaling lines, with a one-dimensional discretization applied to them, thereby reducing computational costs.Then the weighted residual method is applied to obtain the modified SBFEM governing equations and boundary conditions of steady-state seepage problem according to the Laplace diffusion equation and Darcy's law. Furthermore, the steady seepage matrix at infinity is obtained by solving the eigenvalue problem of Schur decomposition and then the 4th-order Runge-Kutta algorithm is used to iteratively solve until the seepage matrix at the boundary lines is reached. Comparisons between the present numerical results and solutions available in the published work have been conducted to demonstrate the efficiency and accuracy of this method. At the same time, the influences of the geometric parameters and complex half-space domain on the seepage flow characteristics in complex half-space domain are investigated in detail.

To evaluate the spatiotemporal evolution of pore water pressure in unsteady seepage flow ahead of a tunnel face, a partial differential equation for unsteady seepage is established. The ranges and boundary conditions of the unsteady seepage flow are specified, and the analytical solution of the unsteady seepage flow is obtained by the eigenfunction method. The obtained analytical solution additionally considers the time factor, which can be used to study the influence of seepage time on the seepage flow. And the pressure transmitting coefficient is introduced to analyze the influence of water and soil characteristics on the unsteady seepage. The analysis shows that the spatiotemporal evolution of the unsteady seepage flow pore water pressure ahead of a tunnel face is reflected in two aspects, the dissipation of the water pressure and the diffusion of the influence range of the unsteady seepage. The dissipation captures the gradual reduction of pore water pressure at a specific location as time progresses. Meanwhile, diffusion characterizes the alteration in the spatial distribution of water pressure. The pressure transmitting coefficient promotes the rate of unsteady seepage, while the height of water table has a greater influence on the magnitude of water pressure change in unsteady seepage flow.

Suction-induced seepage flow can significantly reduce the soil resistance during the installation of suction buckets, thereby ensuring their intended penetration depths and the designed in-service capacities. However, the lack of analytical models describing seepage flow behavior in the literature poses a significant challenge, primarily due to the complexity of seepage boundary conditions and the anisotropy and spatial variation of soil permeability. This paper addresses this gap by presenting a novel analytical solution for analyzing suctioninduced seepage flow around buckets, with a particular focus on the general multilayered soil profile featuring anisotropic permeability. The method of separation of variables is used initially to derive general solutions, and the final solutions are subsequently obtained by combining continuous conditions with the orthogonality of Bessel functions. The accuracy of the solutions is confirmed through comparisons with the results obtained from finite element analysis. Furthermore, this study discusses the seepage flow behavior in several typical scenarios, including permeability anisotropy, increased permeability within the bucket, and the presence of an overlying low-permeability layer. The analytical solutions presented in this paper provide a rapid and accurate method for the analysis of suction-induced seepage flow during suction-assisted installations across a wide range of complex soil permeability conditions.