Two Nitsche-based mixed finite element discretizations for the seepage problem in Richards' equation

TL;DR

This work addresses the challenge of determining seepage faces in Richards' equation for unsaturated groundwater flow by formulating the seepage boundary as a unilateral constraint reminiscent of Signorini/KKT conditions. It introduces two Nitsche-based discretizations on a mass-conserving mixed finite element framework: a non-hybridized penalty approach and a hybridized approach that treats the surface pressure as a separate unknown, both capable of switching between Neumann and Dirichlet regimes automatically. The methods are validated on rectangular, artificial-slope, and real topography domains, showing that the hybridized scheme is more robust to penalty tuning and can mitigate temporal delays through relaxation of seepage constraints. The work advances reliable, mass-conserving simulations of seepage faces and supports coupling with slope-stability analyses for rainfall-induced landslide assessment, with an open-source implementation in PyGeoN.

Abstract

This paper proposes two algorithms to impose seepage boundary conditions in the context of Richards' equation for groundwater flows in unsaturated media. Seepage conditions are non-linear boundary conditions, that can be formulated as a set of unilateral constraints on both the pressure head and the water flux at the ground surface, together with a complementarity condition: these conditions in practice require switching between Neumann and Dirichlet boundary conditions on unknown portions on the boundary. Upon realizing the similarities of these conditions with unilateral contact problems in mechanics, we take inspiration from that literature to propose two approaches: the first method relies on a strongly consistent penalization term, whereas the second one is obtained by an hybridization approach, in which the value of the pressure on the surface is treated as a separate set of unknowns. The flow problem is discretized in mixed form with div-conforming elements so that the water mass is preserved. Numerical experiments show the validity of the proposed strategy in handling the seepage boundary conditions on geometries with increasing complexity.

Figures (16)

• Figure 1: Sketch of the porous domain $\Omega$. The arrows represent the rainfall on the ground surface, that is modeled as a vector field $\mathbf{p}$.
• Figure 2: Left panel shows of the permeability curve $K(\psi)$ for the various soil types. Right panel reports of the water content curve $\theta(\psi)$ for the various soil types.
• Figure 3: Rectangular domain. Test with $p/K_S=0.1$. Top row: comparison between the Neumann, no-hybridized and hybridized scheme's solutions at time instants $0$h, $10$h, $20$h, …, $90$h, $100$h. Bottom row: sensitivity analysis on the numerical parameter $\gamma_0$ for the no-hybridized scheme on the simulation's result at final time.
• Figure 4: Rectangular domain. Test with $p/K_S=1$. Top row: comparison between the Neumann, no-hybridized and hybridized scheme's solutions at time instants $0$h, $5$h, $10$h, …, $45$h, $50$h. Bottom row: sensitivity analysis on the numerical parameter $\gamma_0$ for the no-hybridized scheme on the simulation's result at final time.
• Figure 5: Rectangular domain. Test with $p/K_S=1$. Left panel: number of iterations. Right panel: ratio between the number of nonlinear iterations performed by the Neumann approximation over the number required by the no-hybridized and hybridized scheme.