Using complementarity framework to couple subsurface flow and seepage processes: a physically based basis to integrate hotspots reactivity at the hillslope scale

Abstract : Landscape structure and geological heterogeneities are major controls on subsurface flow dynamic. Particularly it has strong impact on saturated areas (potential hotspots) emergence, promoting seepage production. This control leads to highly non linear flow response where, for each hillslope location, two possible states can be distinguished: with or without seepage. Different algorithmic solutions have been proposed to model this process. A solution is to solve subsurface flow for an assumed water table position and then iterate on the water table position until convergence is met. Seepage areas and seepage values are then deduced from the locations where water table intersects surface. A second way to proceed is to explicitly couple groundwater equations and surface water equations with an exchange flux. Here we developed a novel approach using the complementarity framework to reconcile in a single system the two states potentially encountered (with or without seepage). The complementarity framework manages the current state and the possible transition between states thanks to a specifically devoted equation. This framework is applied to the 1D hillslope storage Boussinesq equations (Troch et al. 2003). Reformulating this complementarity system differently enables to partition effectively the local flux balance between storage variation and seepage. This differential algebraic equations (DAEs) system has the major benefit to be directly solvable with built-in ode libraries. Finally the system is regularized to enhance fast and efficient solving. This model is stable with fast spatial convergence. It respects the mass balance locally beyond the tolerance limits and shows limited sensitivity to the value of the regularization parameter. The model appears to be robust, able to solve complex realistic case with presence of landscape heterogeneity and real hydrologic forcing. This model will then be used as a physical basis to implement biogeochemistry reactivity.