Finite element formulation of unilateral boundary conditions for unsaturated flow in porous continua

This paper presents the numerical resolution of unilateral boundary conditions able to effectively model several problems of unsaturated flow, as those involving rainfall infiltration and seepage faces. Besides the penalty technique, we also consider the novel regularization of these conditions by means of the more effective augmented Lagrangian method. The performance of the so-obtained finite element method is carefully investigated in terms of accuracy and ill-conditioning effects, including comparisons with analytical solutions and a complete identification of the analogies with the problem of frictionless contact. In this way, we provide a priori estimates of optimal and admissible ranges for the penalty coefficient as functions of permeability and spatial discretization. The proposed method and the estimated coefficient ranges are validated in further numerical examples, involving the propagation of a wetting front due to rainfall and the partial saturation of an aged concrete dam. These applications show that the proposed regularizations do not induce any detrimental effect on solution accuracy and on convergence rate of the employed Newton-Raphson method. Hence, the present approach should be preferred to the commonly used iterative switching between the imposed-flow and the imposed-pressure conditions, which often leads to spurious oscillations and convergence failures.