A robust solution to Richards' equation with use cases in complex soil hydraulic models, non-equilibrium unsaturated flow in soil and model coupling using the Method Of Lines

Modeling soil hydraulic processes requires robust and stable numerical solutions, also when computational resources are limited. Different challenging problems like sudden changes of pressure or fluxes at the boundary of the model domain or very dry initial conditions are challenges for standard numerical solution methods such low-order finite difference and finite element methods. The Method of Lines approach is proven to achieve numerical robustness and stability while allowing the handling of different complex soil hydraulic models for one-dimensional problems. To be applicable in a wide range of scenarios the method should also be easily extensible. Here the Method Of Lines approach is shown to enable the handling of different complex soil hydraulic models, the modification of Richards' equation to consider non-equilibrium effects and the extension with a lateral flow model to form a combined 1.5D hillslope model.

 

A slightly modified Method of Lines approach is used to solve the pressure based 1D Richards' equation. A finite differencing scheme is applied to the spatial derivative and the resulting system of ordinary differential equations is reformulated as differential-algebraic system of equations. The open-source code IDAS from the Sundials suite is used to solve the DAE system. To show the broad applicability of the method, several successful use cases are presented. These range from the inclusion of more complex soil hydraulic models to be able to consider hystersis effects and dual-permeability flow over the extension of Richards' equation to model non-equilibrium unsaturated flow to linking the Richards' equation with the Boussinesq lateral flow equation to form an efficient 1.5-D hillslope model.

 

The results show that the Method of Lines approach for solving Richards' equation satisfies the required conditions of numerical robustness and stability and allows for easily including new processes and a wider set of applications.