  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.
