A general SPH framework for transient seepage flows through unsaturated porous media considering anisotropic diffusion

Abstract

The Smoothed Particle Hydrodynamics (SPH) method has become a popular numerical tool to solve many challenging engineering applications. However, its application to solving anisotropic transient unsaturated–saturated seepage flow problems is still absent in the literature. Among several challenges involved in solving this important problem, the lack of a robust and general SPH formulation to obtain accurate approximations of second derivatives on disordered particle system is the key obstacle preventing this to happen. In this paper, we first propose a general SPH formulation for the second derivatives, capable of considering anisotropic diffusions, and demonstrate that the new formulation outperforms existing SPH formulations for the second derivatives and achieves high accuracy on the highly disordered particle systems. Subsequently, we utilise this newly developed SPH formulation to tackle the transient seepage flows through porous media for the first time, considering anisotropic flows and complete time-dependent transition from unsaturated to saturated states and vice-versa. This naturally leads to a general SPH framework for solving transient seepage problems in unsaturated/saturated porous media using one set of Lagrangian particles (or Lagrangian discretisation). Robust boundary conditions and their implementations for solving general seepage problems in SPH are also proposed, enabling the proposed SPH model to automatically capture seepage surface (including the free surface of unconfined seepage) without any difficulty or ad hoc treatment. Moreover, a simple method to generate a random particle system suitable for SPH simulations on large scale problems using the Voronoi tessellation technique is proposed to study transient seepage problems for the first time. Several verification examples are presented to demonstrate the capabilities of the proposed SPH framework in solving the complex seepage flow problems, including those on multi-resolution particle configurations. Excellent agreements with a wide range of numerical and theoretical solutions are achieved, suggesting that the newly proposed SPH model can be applied to solve complex transient seepage flow problems. This marks an important milestone for the future development of SPH to solve fully coupled flow deformation problems involved large deformation and failure of porous media, which are challenging for the most state-of-the-art mesh-based numerical methods.