  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 don't require mesh generation. Furthermore,
these 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.
