A numerical formulation with unified unilateral boundary condition for unsaturated flow problems in porous media

Abstract

This paper proposed a numerical formulation for unsaturated flow problems with nonlinear boundaries of seepage face and soil–atmosphere interface via the concept of parabolic variational inequality (PVI) method. A unified unilateral boundary condition was first proposed to represent the conditions on the seepage face and soil–atmosphere interface boundaries within the partial differential equation (PDE) formulation. A PVI formulation mathematically equivalent to the PDE formulation was then proposed, which automatically transforms the flux part of the unified unilateral boundary condition into the natural boundary condition and eliminates the singularity at seepage points. By discretizing the PVI formulation, a finite element procedure together with an iterative algorithm was suggested. An existing experiment of unsaturated flow in a layered hillside and a laboratory test of unsaturated flow through sand flume performed in this study were used to validate the proposed method, with a good agreement between the measured and computed results and a satisfactory balance of mass being maintained during the simulations. The numerical results also indicated that the problem of mesh dependence associated with unsaturated flow simulations is well addressed with the proposed numerical method. Finally, the process of unsaturated flow in a soil slope with layers of horizontal drains subjected to rainfall/evaporation was further examined. The numerical results reveal that the deployment of drains in a soil slope can significantly lower the pore water pressure around the drains, with the bottom layer drains being most effective in controlling the seepage flow.