Implicit EXP-RBF techniques for modeling unsaturated flow through soils with water uptake by plant roots

TL;DR

This work addresses modeling unsaturated flow in soils with plant root water uptake by solving the Richards equation with a sink term using a meshfree, implicit EXP-RBF spatial discretization combined with a second-order backward differentiation formula for time stepping. The method employs a modified Picard linearization and yields a sparse system, avoiding mesh generation and mitigating ill-conditioning. Validation against analytical solutions and Hydrus demonstrates accurate soil moisture and pressure-head predictions and good mass conservation, along with notable computational efficiency gains over lower-order temporal schemes. The approach offers a practical, scalable tool for simulating soil-water-plant interactions in hydrology and agricultural management.

Abstract

Modeling unsaturated flow through soils with water uptake by plan root has many applications in agriculture and water resources management. In this study, our aim is to develop efficient numerical techniques for solving the Richards equation with a sink term due to plant root water uptake. The Feddes model is used for water absorption by plant roots, and the van-Genuchten model is employed for capillary pressure. We introduce a numerical approach that combines the localized exponential radial basis function (EXP-RBF) method for space and the second-order backward differentiation formula (BDF2) for temporal discretization. The localized RBF methods eliminate the need for mesh generation and avoid ill-conditioning problems. This approach yields a sparse matrix for the global system, optimizing memory usage and computational time. The proposed implicit EXP-RBF techniques have advantages in terms of accuracy and computational efficiency thanks to the use of BDF2 and the localized RBF method. Modified Picards iteration method for the mixed form of the Richards equation is employed to linearize the system. Various numerical experiments are conducted to validate the proposed numerical model of infiltration with plant root water absorption. The obtained results conclusively demonstrate the effectiveness of the proposed numerical model in accurately predicting soil moisture dynamics under water uptake by plant roots. The proposed numerical techniques can be incorporated in the numerical models where unsaturated flows and water uptake by plant roots are involved such as in hydrology, agriculture, and water management.

Figures (15)

• Figure 1: Schematic of the plant water stress response function FeddesR.A1978Sofw.
• Figure 2: Example of influence domain $\Omega_s$ with seven points within a cube.
• Figure 3: Case 1: $\alpha=0.01~cm^{-1}$ and $R_0=0.02~h^{-1}$. Comparison between approximate and exact solutions for water content and pressure head. Left: with root water uptake. Right: without root water uptake.
• Figure 4: Case 1: $\alpha=0.01~cm^{-1}$ and $R_0=0.02~h^{-1}$. Comparison between approximate and exact solutions for water content and pressure head. Left: with root water uptake. Right: without root water uptake.
• Figure 5: Root mean square error (RMSE) as a function of the time step for BDF1 and BDF2 schemes. Left: $\alpha=0.1~cm^{-1}$ and $R_0=0.0025~h^{-1}$. Right: $\alpha=0.01~cm^{-1}$ and $R_0=0.02~h^{-1}$.