Stochastic Differential Equations to Decide Soil Water Movement.

Soil water content influences soil physical, chemical, and biological processes, and it is a vital factor in agricultural production. Stochastic methods, such as auto-regressions, moving average models and Kalman filters, are used to predict changes in soil water content. However, such methods are semi-empirical and are limited by measurement resolution. A stochastic Richards equation may be used to simulate soil water movement and determine soil water content distributions, when the hydraulic conductivity is considered to be a random field.

The objective for this study is to solve the stochastic Richards equation and improve the numerical scheme. A Karhunen-Loeve (KL) polynomial expansion is used to construct a lognormal random field for hydraulic conductivity, and a multi-scale finite element method (MSFEM) is used to solve the stochastic Richards equation. Triangular elements are used to construct the numerical MSFEM basis functions, and an oversampling method is used to adjust the error that occurs at the boundary regions of the basis functions. A finite difference method (FDM) is used to update the numerical MSFEM basis functions. Simulation examples show that KL polynomial expansion is an effective method for generating the random field, the oversampling method can improve the accuracy of basis functions, and the MSFEM can provide valid simulation results for soil water movement. Thus, the MSFEM scheme is an easy and effective way to predict soil water content distributions.