(356b) A Data-Driven Numerical Method for Solving Water Flow Dynamics in Soil

Modeling and predicting soil moisture is essential for precision agriculture, smart irrigation, drought prevention, etc. Estimating root zone soil moisture from the surface or near-surface soil moisture data is typically achieved by solving a hydrological model that describes water movement through soils. Most existing agro-hydrological models are based on the Richards equation [1], which captures irrigation, precipitation, evapotranspiration, runoff, and drainage dynamics in soil. The Richards equation is a highly nonlinear, degenerate elliptic-parabolic partial differential equation (PDE) that is notoriously difficult to solve. State-of-the-art Richards equation solvers rely on spatial discretization based on finite difference, finite element, or finite volume method (FVM). Unlike finite difference and finite element methods, FVM adopts an integral form of the Richards equation, which offers some valuable physical insights (e.g., mass conservation) about water flow dynamics. However, these insights are often lost again in the process of transforming the discretized equations into a matrix equation. Furthermore, the resulting matrix is stiff and sparse and is computationally challenging to solve. To address these numerical difficulties, several linear iterative schemes have been proposed, among which the L-scheme exhibits the nice property of unconditional convergence which is irrespective of the choice of initial guess [2,3]. Nevertheless, in the standard L-scheme, the linearization parameter is an arbitrarily chosen static constant that is not adaptive to any change in pressure head. This greatly impacts the performance of the L-scheme. In the first part of this talk, we introduce a novel adaptive L-scheme whose linearization parameters dynamically adjust themselves for each discretized cell, time step, and iteration, thereby significantly improving the stability and robustness of the L-scheme.

To further incorporate the underlying physics of water flow dynamics and conservation in soil, in the second part of this talk, we introduce the first data-driven global random walk algorithm to solve the FVM-based adaptive L-scheme. A key assumption made in existing global random walk algorithms for solving the Richards equation is that the pressure head is proportional to the number of particles in a discretized cell [4]. Nevertheless, we have shown that this assumption is invalid, and the relationship between the pressure head and the number of particles may not be continuous, smooth, or explicit. Instead, we propose a novel data-driven approach and used two neural networks to accurately learn the mapping and inverse mapping between the pressure head and the number of particles. Coupling this with the adaptive L-scheme, we show that our data-driven framework not only is the first-of-its-kind that can solve 3-D Richards equation, but also outperformed commercial and state-of-the-art solvers in accurately capturing the underlying physics.

References

[1] L.A. Richards, Capillary conduction of liquids through porous mediums, Physics, 1931, 1(5): 318-333.

[2] K. Mitra, I. Pop, A modified l-scheme to solve nonlinear diffusion problems, Computers & Mathematics with Applications, 2019, 77(6): 1722-1738.

[3] F. List, F. Radu, A study on iterative methods for solving Richard’s equation, Computational Geoscience, 2016, 20: 341-353.

[4] N. Suciu, D. Illiano, A. Prechtel, F. A. Radu, Global random walk solvers for fully coupled flow and transport in saturated/unsaturated porous media, Advances in Water Resources, 2021, 152: 103935.