A semi-implicit finite element method for modeling fertilizer transport through soils with root water and nutrient uptake

Modeling the interactions of water flow, fertilizer, and root water and nutrient uptake in soils is crucial for optimizing water and nutrient management in agriculture. In this study, we propose a new semi-implicit finite element method for modeling fertilizer transport in unsaturated porous media. The numerical model incorporates root water and nutrient uptake by plants, considering passive, active, and combined passive–active uptake mechanisms. These processes add nonlinear terms that can lead to numerical issues which are addressed through the proposed semi-implicit treatments based on second-order Taylor approximation techniques. The proposed approach linearizes the system of governing partial differential equations in time. The effectiveness of the scheme is tested using numerical experiments. First, the coupled model of infiltration and solute transport in unsaturated zone connected to a groundwater table is simulated. Next, the transport of a urea–ammonium–nitrate fertilizer mixed with phosphorus and potassium fertilizers through the soil with subsurface drip fertigation is modeled where nitrogen transformations such as hydrolysis and nitrification are included. The performance of the proposed scheme is evaluated by comparing the results with those obtained using the well-established Hydrus software based on implicit methods. The numerical experiments demonstrate the robustness and accuracy of the proposed semi-implicit scheme in handling nonlinearities of the coupled system. These results demonstrate that the proposed approach offers an efficient alternative to standard implicit methods for modeling water and nutrient dynamics in unsaturated porous media.