A semi-implicit second-order temporal scheme for solving the pressure head-based form of Richards' and advection-dispersion equations

In this study, a semi-implicit finite element method is proposed to solve the coupled system of infiltration and solute transport in porous media. The Richards equation is used to describe unsaturated flow, while the advection-dispersion equation (ADE) is used for modeling solute transport. The proposed approach is applied to linearize the system in time, avoiding iterative processes. A free parameter is introduced to ensure the stability of the scheme. Numerical tests are conducted to analyze the accuracy of the proposed method in comparison with three second-order iterative schemes. The proposed scheme based on the optimal free parameter is accurate and efficient since it offers a considerable gain in computational time compared to the other methods. For reliability and effectiveness evaluation of the developed semi-implicit scheme, four showcase scenarios are used. The first two numerical tests focus on modeling water flow in heterogeneous medium and transient flow in variably saturated zones. The last numerical tests are carried out to simulate the salt and nitrate transport through unsaturated porous media. The simulation results are compared with reference solutions and laboratory data, and demonstrate the effectiveness of the proposed scheme in simulating infiltration and solute transport in porous media.