Using H‐Convergence to Calculate the Numerical Errors for 1D Unsaturated Seepage in Transient Conditions

Abstract

Numerical modeling plays a pivotal role in understanding transient unsaturated flow, which is critical for applications such as groundwater recharge, stormwater management, and contaminant transport. This study investigates the effect of time step refinement on numerical solutions for a vertical infiltration 1D test in a vertical column. First, the element size, ES, was selected to have all calculations in the MCD, the mathematical convergence domain. Then, the numerical and mathematical convergences of the numerical solutions were studied versus the time step, Δt. The study provided results for hydraulic head, unsaturated hydraulic conductivity, volumetric water content, and vertical water velocity versus elevation, elapsed time t, and Δt. Asymptotic behavior was obtained for all unknowns when Δt → 0. All results for all parameters gave linear relationships between the log of the numerical error and the log of Δt, as predicted by mathematics. Thus, they proved that a good code converges mathematically when ES and Δt are decreased. For this 1D problem, the MCD is reached for Δt ≤ 1 s, which is a small MCD. For larger Δt values (2 ≤ Δt ≤ 100 s), the code converges numerically in the numerical convergence domain (NCD), where the solutions respect the user‐defined convergence criteria but deviate from the true mathematical convergence criterion, underscoring that a fine temporal discretization is critical. The study also demonstrates that small Δt steps ensure physically consistent behavior, as reflected in smooth slope volumetric water content versus suction curves, whereas large Δt steps produce oscillations and instability.