Simulation of saturated–unsaturated seepage problems via the virtual element method

The virtual element method (VEM) demonstrates excellent convergence when dealing with general polygonal mesh shapes and addressing nonlinear problems. Saturated–unsaturated seepage represents a highly nonlinear problem, and generating appropriate meshes poses a challenge in seepage studies. This paper utilizes the VEM to discretize the Richards equation by employing the hydraulic head formulation and introduces computational methods for the corresponding stiffness matrix and mass matrix. Five numerical examples are employed to validate the effectiveness and feasibility of the algorithm in solving saturated–unsaturated seepage problems. In the verification part, the accuracy of the algorithm and the convergence of the mesh shape are verified by comparing the analytical solution of the saturation-unsaturated seepage problem in homogeneous soil with the classical experimental results. Then, the effectiveness of the proposed algorithm in solving the saturation-unsaturated seepage problem of layered soil is verified by comparing with the analytical solution and other research results. Finally, an engineering example is given to illustrate the feasibility of the algorithm for solving engineering problems. The simulation results demonstrate that the VEM can effectively simulate saturated–unsaturated seepage in homogeneous and layered soils and exhibits strong convergence.