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

TL;DR

This work tackles the numerical solution of coupled infiltration and solute transport in unsaturated soils by solving the pressure-head-based Richards equation alongside the ADE. It develops a semi-implicit, second-order temporal scheme (SILF2) with finite element space discretization and a stabilizing parameter $\nu$, enabling a noniterative linear solve while preserving accuracy. Through comprehensive tests including Green–Ampt infiltration, VG- and Gardner-based soils, and comparisons with analytical, experimental, and Hydrus data, SILF2 with the optimal $\nu=1$ demonstrates superior efficiency and robustness. The method holds practical significance for reliable, fast simulations of infiltration and contaminant transport in variably saturated soils, with broad applicability to irrigation, hydrology, and environmental studies.

Abstract

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.

Figures (19)

• Figure 1: Flowchart diagram for methodology implementation.
• Figure 2: Schematic illustration of the two-dimensional infiltration tests.
• Figure 3: The boundary condition enforced on the top of the soil. (a) and (b) refer to Test 1 and Test 2, respectively.
• Figure 4: $L^2$ error on log-scaled of the pressure head for Test 1 (left column) and Test 2 (right column) as a function of time step (top), mesh size (middle), and CPU time (bottom) for all proposed methods.
• Figure 5: $L^2$-error for Test 1 and Test 2 on log-log scale using the SILF2 method with different values of the free parameter $\nu$.

Theorems & Definitions (4)

• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4