(241e) A Computationally Efficient Data-Driven Framework for Solving Water Flow Dynamics in Soil Via Fractional Diffusion Model

Modeling and predicting soil moisture is essential for precision agriculture, smart irrigation, drought prevention, etc. Estimating root zone soil moisture from 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 characterizes irrigation, precipitation, evapotranspiration, runoff, and drainage dynamics in soil as a nonlinear diffusion process. The classical Richards equation, which combines Darcy’s law and continuity equation, is a first-order nonlinear degenerate elliptic-parabolic partial differential equation that is notoriously challenging to solve. Furthermore, recent experimental studies show that water flow dynamics in soil exhibit anomalous non-Boltzmann scaling behavior. This suggests generalizing the classical Richards equation by introducing fractional time derivatives, thereby making the resulting fractional Richards equation even more difficult to solve efficiently. While finite difference and finite point methods are generally used to discretize the fractional time derivatives, their computational costs are often high as the discretized equations are solved as a large-scale sparse matrix equation. Furthermore, valuable physical insights (e.g., mass conservation) about water flow dynamics will be lost during this process.

In this talk, we present a novel finite volume discretization-based numerical scheme that simultaneously incorporates L₁ interpolation, adaptive L-scheme, global random walk, and neural network to efficiently solve 3-D fractional Richards equation for the first time. Specifically, we use L₁ interpolation to discretize the Caputo fractional derivatives and develop an adaptive linearization procedure to solve the discretized Richards equation iteratively. Unlike the standard L-scheme in which the linearization parameter is an arbitrarily chosen static constant, in our adaptive L-scheme [3,4], the linearization parameters can 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, we develop a data-driven global random walk algorithm and incorporate it in the discretization framework. A key assumption made in existing global random walk algorithms for solving the classical Richards equation is that the pressure head is proportional to the number of particles in a discretized cell [5]. However, 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. Thus, our novel data-driven approach uses two neural networks to accurately learn the mapping and inverse mapping between the pressure head and the number of particles. Coupling this with our discretization framework, in an illustrative case study, we show that our numerical scheme not only successfully solves 3-D fractional Richards equation for the first time, but also accurately preserves the relationships between water flux, soil moisture, and pressure head.

References

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

[2] E. Gerolymatou, I. Vardoulakis, R. Hilfer, Modelling infiltration by means of a nonlinear fractional diffusion model, 2006, Journal of Physics D: Applied Physics, 39(18): 4104.

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

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

[5] 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.