A perturbative view on the subsurface water pressure response at hillslope scale

Abstract

This work deals with a rearrangement of Richards Equation to obtain a better understanding of the components of the subsurface water and pressure flows and their interactions in a hillslope. The basic idea proposed in this paper is the normalization of the spatial and temporal coordinates according to the physical knowledge of the processes and the geometry of a hillslope. The pressure head and all the other quantities contained in Richards Equation are rescaled to obtain a dimensionless equation. The pressure head is also split into an equilibrium component and a transient component and expanded in a perturbation series. Subsequently, Richards Equation itself is conveniently split into two equations connected by a source/sink term. Results of these manipulations show that, for the first order approximation, the water and pressure flow is slope‐normal, while slope‐parallel effects only occur at successive orders. Integrating the perturbed equation obtained for the long‐term pressure head component results in the well‐known Boussinesq Equation for the water table level. Further integrations then lead to the hillslope‐storage Boussinesq model and finally to a generalization of the O'Loughlin formula for the slope‐parallel long‐term subsurface flow. The introduced source/sink term is estimated at the first order of approximation to be proportional to the temporal variation of water pressure head at the bottom of the integration domain. The treatment of the equations clarifies some of the mechanisms acting in water pressure redistribution at hillslope scale, even without solving the equation.