A similarity framework to assess controls on shallow subsurface flow dynamics in hillslopes

An understanding of the dominant controls on water fluxes through landscapes at scales smaller than the resolution of a hydrologic model is useful for choosing appropriate parameterizations, and for developing a classification system for landscape hydrology. The links between these controls and the parameters in hillslope-scale models of subsurface lateral saturated flow have been treated implicitly in both physical (e. g., Characteristic Response Function) and empirical formulations, such as power law storage-discharge relations. In particular, the controls exerted by the boundary condition at the toe of the hilslope and the temporal variability of the recharge have not been integrated with the understanding of the controls of topography and soil properties. In this work, we develop a dimensionless similarity framework for assessing these controls on the subsurface flow dynamics, based on the Boussinesq equation in an idealized hillslope system. Using this framework we demonstrate the relationship between (1) the exponents of the storage-discharge relations, (2) the boundary conditions and (3) the characteristic storage thickness used in the Characteristic Response Function parameterization. We use the concept of hydrologic regimes to incorporate information about the population of recharge events that drive the system and set up the initial condition of saturated storage for each storm event. The analysis demonstrates that for planar hillslopes with uniform soil properties, exponents range between 0 (when the flow through the hillslope is dominated by the topographic gradients, and is in the early stages of draining) and 2 (when the flow is driven by water table gradients, and the thickness of the aquifer at the seepage face is small compared to the average thickness). The similarity framework can be used to make first-order predictions of these exponents, and classify hillslopes using a simple typology of subsurface flows based on two components of the behavior: the relative dominance of the topographic and water-table gradients, and the responsiveness of the hillslope discharge to temporal variability of the recharge inputs.