A finite element method using a bounded auxiliary variable for solving the Richards equation

TL;DR

This work advances infiltration modeling in porous media by solving the Richards equation with a bounded auxiliary variable $u$ that eliminates unbounded terms in the weak form. A semi-implicit finite element scheme paired with Newton iterations is developed, and heterogeneity is tackled via a non overlapping Schwarz domain decomposition. The approach avoids regularization and variable switching, remaining well defined for both dry and saturated states, and is validated across one and two dimensional tests including layered and heterogeneous soils. Results demonstrate stability, positivity of solutions, and accuracy consistent with manufactured solutions and established benchmarks, highlighting practical applicability to unsaturated flow problems in complex media.

Abstract

The Richards equation, a nonlinear elliptic parabolic equation, is widely used to model infiltration in porous media. We develop a finite element method for solving the Richards equation by introducing a new bounded auxiliary variable to eliminate unbounded terms in the weak formulation of the method. This formulation is discretized using a semi-implicit scheme and the resulting nonlinear system is solved using Newton's method. Our approach eliminates the need of regularization techniques and offers advantages in handling both dry and fully saturated zones. In the proposed techniques, a non-overlapping Schwarz domain decomposition method is used for modeling infiltration in layered soils. We apply the proposed method to solve the Richards equation using the Havercamp and van Genuchten models for the capillary pressure. Numerical experiments are performed to validate the proposed approach, including tests such as modeling flows in fibrous sheets where the initial medium is totally dry, two cases with fully saturated and dry regions, and an infiltration problem in layered soils. The numerical results demonstrate the stability and accuracy of the proposed numerical method. The numerical solutions remain positive in the presence of totally dry zones. The numerical investigations clearly demonstrated the capability of the proposed method to effectively predict the dynamics of flows in unsaturated soils.

Figures (21)

• Figure 1: The absolute value of the Leverett function for various capillary pressure models.
• Figure 2: Graph of $u$ as a function of $S$.
• Figure 3: The boundary conditions.
• Figure 4: Layered soil problem.
• Figure 5: 1D boundary conditions.