Use of generalized power series for the modeling of 1D and 3D (disc-source) water infiltration into soils

Climate and global changes increase pressure on natural resources, particularly water. Climate change affects hydrological processes and threatens water resources in both quantity and quality. Societal adaptation therefore requires a paradigm shift in water management, including reducing human impacts on the water cycle and restoring the natural cycle. Achieving this transition relies on a detailed understanding of hydrological processes, especially those governing water infiltration into soils. Soil water infiltration modeling has been studied for decades, with approaches ranging from analytical to numerical modeling. Analytical solutions, developed as approximations of Richards’ equation, were initially favored before advances in computational capacity enabled numerical models. Despite this evolution, analytical approaches remain essential for validating and consolidating numerical developments. Among the analytical models proposed, power series expansions in time—more precisely in t^1/2 - were the earliest, based on Philip’s pioneering work (Philip, 1957). Later, Haverkamp et al. (1994) introduced an implicit quasi-exact formulation for infiltration into soils with uniform initial water content, afterwards adapted to circular surface sources. These models, along with their short-time expansions, form the basis for experimental data analysis and are typically truncated after the first three terms, as higher-order contributions are negligible (Moret-Fernández et al., 2020).

This study investigates a general power series formulation for modeling water infiltration, I(t) = a₁ t^α1 + a₂ t^α2 + a₃ t^α3, and evaluates its ability to fit numerically generated infiltration data for different choices of exponents and coefficients. The study first demonstrates that the simultaneous estimation of all three exponents and coefficients leads to an ill-posed inversion problem due to model overparameterization. The analysis is therefore restricted to a two-term formulation, I(t) = a₁ t^α1 + a₂ t^α2, with parameters optimized sequentially to reduce non-uniqueness. One-dimensional horizontal infiltration data are first analyzed using the single-term model I(t) = a₁ t^α1, with several inversion strategies, including fixing the parameters to the reference values a₁ = S (soil sorptivity) and α₁=1/2. One-dimensional vertical infiltration and disc-source infiltration are then fitted to estimate the remaining parameters a₂ and α₂. Parameter estimation options, including reference values from Haverkamp et al. (1994), are evaluated across multiple soils and initial saturation conditions. Finally, the obtained parameter values are discussed in light of physical considerations. This study aims to contribute to the development and application of analytical approaches for modeling water infiltration.

References

• Haverkamp, R., Ross, P.J., Smettem, K.R.J., Parlange, J.Y., 1994. Three-dimensional analysis of infiltration from the disc infiltrometer. 2. Physically based infiltration equation. Water Resour. Res. 30, 2931–2935.
• Moret-Fernández, D., Latorre, B., López, M.V., Pueyo, Y., Lassabatere, L., Angulo-Jaramilo, R., Rahmati, M., Tormo, J., Nicolau, J.M., 2020. Three- and four-term approximate expansions of the Haverkamp formulation to estimate soil hydraulic properties from disc infiltrometer measurements. Hydrol. Process. 34, 5543–5556. https://doi.org/10.1002/hyp.13966
• Philip, J., 1957. The theory of infiltration: 1. The infiltration equation and its solution. Soil Sci. 83, 345–358.

This project has received funding from European Union’s HORIZON EUROPE research and innovation program GA N°101072777-PlasticUnderground HEUR-MSCA-2021-DN-01.