Localized RBF methods for modeling infiltration using the Kirchhoff-transformed Richards equation

We develop a new approach to solve the nonlinear Richards equation based on the Kirchhoff transformation and localized radial basis function (LRBF) techniques. Our aim is to reduce the nonlinearity of the governing equation and apply LRBF methods for modeling unsaturated flow through heterogeneous soils. In our methodology, we propose special techniques which deal with the heterogeneity of the medium in order to apply the Kirchhoff transformation where we used the Brooks and Corey model for the capillary pressure function and a power-law relation in saturation for the relative permeability function. The new approach allows us to avoid the technical issues encountered in the Kirchhoff transformation due to soil heterogeneity in order to reduce the nonlinearity of the model equation. The resulting Kirchhoff-transformed Richards equation is solved using LRBF methods which have advantages in terms of computational cost since they do not require mesh generation. Furthermore, LRBF techniques lead to a system with a sparse matrix which allows us to avoid ill-conditioned issues. To validate the developed approach for predicting the dynamics of unsaturated flow in porous media, numerical experiments are performed in one, two, and three-dimensional soils. The numerical results demonstrate the efficiency and accuracy of the proposed techniques for modeling infiltration through heterogeneous soils.

Automated summary

We propose a new approach to solve the nonlinear Richards equation using the Kirchhoff transformation and localized radial basis function (LRBF) techniques. This method reduces nonlinearity and models unsaturated flow through heterogeneous soils. We introduce special techniques to handle medium heterogeneity and apply the Kirchhoff transformation with the Brooks and Corey model and a power-law relation. The resulting equation is solved using LRBF methods, which are computationally efficient and avoid mesh generation.