In this study, a novel semi-implicit second-order temporal scheme combined with the finite element method for space discretization is proposed to solve the coupled system of infiltration and solute transport in unsaturated 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 used to linearize the system of equations in time, eliminating the need of 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 a family of second-order iterative schemes. The proposed numerical technique based on the optimal free parameter is accurate and performs better in terms of efficiency 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 soil and transient flow in variably saturated zones. The last numerical tests are carried out to simulate the salt and nitrate transport through unsaturated soils. 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 through unsaturated soils.

翻译：本研究提出了一种结合空间离散有限元方法的新型半隐式二阶时间格式，用于求解非饱和多孔介质中入渗与溶质运移的耦合系统。采用Richards方程描述非饱和流动，对流-弥散方程（ADE）模拟溶质运移。所提方法对时间方向进行线性化处理，无需迭代过程。引入自由参数确保格式稳定性。通过数值试验分析所提方法与系列二阶迭代格式的精度差异。基于最优自由参数的数值技术具有高精度且效率更优，相比其他方法可显著节省计算时间。为评估该半隐式格式的可靠性与有效性，设计了四个算例场景：前两个数值试验聚焦于非均质土壤水流模拟及变饱和区域瞬态流动；后两个试验则模拟非饱和土壤中盐分与硝酸盐的运移过程。仿真结果与参考解及实验室数据对比表明，该格式在模拟非饱和土壤入渗与溶质运移方面具有显著有效性。