Numerical models using finite element techniques for water dynamics and transport of fertilizers in unsaturated soils

Abstract

Global agricultural systems face the dual challenge of increasing food production to meet growing population demands while minimizing environmental impacts from water and fertilizer use. Understanding and predicting water infiltration and nutrient transport in unsaturated soils is fundamental to addressing this challenge, as these coupled processes control crop water availability, fertilizer use efficiency, and the risk of groundwater contamination. Despite their importance, these processes remain difficult to model accurately due to their highly nonlinear nature, sharp moving fronts, complex coupling between flow and transport phenomena, and unstable infiltration patterns such as gravity fingering in dry sandy soils. This thesis develops advanced numerical methodologies to address these challenges, based on the Richards equation for variably saturated water flow and the advection-dispersion equation for solute transport. The models also include root water and nutrient uptake mechanisms and nitrogen transformation processes. The proposed numerical techniques employ various strategies. First, semi-implicit and predictor corrector temporal discretization schemes in conjunction with standard and mixed f inite element methods in space are developed to eliminate iterative nonlinear solvers while maintaining high-order accuracy and numerical stability for the coupled system of equations. These approaches employ sophisticated linearization techniques including Taylor expansion and temporal extrapolation to handle nonlinearities arising from hydraulic properties, root water and nutrient uptake. Second, computational efficiency is achieved through sophisticated parallel computing tools using the JAX library, combining GPU acceleration with just-in-time compilation, automatic differentiation, and vectorized operations to achieve substantial performance improvements. Third, the modeling framework is extended beyond the classical Richards equation to address unstable infiltration through the BCJ-R equation, employing mixed finite element methods with novel linearization strategies. The framework is also integrated into the JAX ecosystem to accelerate computational time. The developed methodologies are validated through comprehensive numerical experiments including comparisons with analytical solutions, established software packages, and experimental data. Applications encompass heterogeneous media simulations, transient variably saturated flow, salt and nitrate transport under surface and subsurface drip irrigation, and complex multi-nutrient fertilizer dynamics with nitrogen transformation processes. Performance analyses demonstrate substantial computational efficiency improvements, with GPU implementations achieving significant speedups compared to serial CPU codes for large-scale problems. Simulations of unstable infiltration successfully reproduce experimental observations including saturation overshoot and two-dimensional fingering patterns across different flux conditions. The contributions of this thesis advance the state-of-the-art in computational subsurface hydrology by providing efficient, robust, and accurate tools for simulating coupled water and nutrient dynamics in agricultural systems.

Date	1 Apr 2026
Original language	American English
Awarding Institution	École de technologie supérieure
Supervisor	Abdelaziz Beljadid (Supervisor) & Azzeddine Soulaïmani (Co-supervisor)