Based on a literature overview, this paper summarizes the impact and legacy of the contributions of Wilfried Brutsaert and Jean-Yves Parlange (Cornell University) with respect to the current state-of-the-art understanding in hydraulic groundwater theory. Forming the basis of many applications in catchment hydrology, ranging from drought flow analysis to surface water-groundwater interactions, hydraulic groundwater theory simplifies the description of water flow in unconfined riparian and perched aquifers through assumptions attributed to Dupuit and Forchheimer. Boussinesq (1877) derived a general equation to study flow dynamics of unconfined aquifers in uniformly sloping hillslopes, resulting in a remarkably accurate and applicable family of results, though often challenging to solve due to its nonlinear form. Under certain conditions, the Boussinesq equation can be solved analytically allowing compact representation of soil and geomorphological controls on unconfined aquifer storage and release dynamics. The Boussinesq equation has been extended to account for flow divergence/convergence as well as for nonuniform bedrock slope (concave/convex). The extended Boussinesq equation has been favorably compared to numerical solutions of the three-dimensional Richards equation, confirming its validity under certain geometric conditions. Analytical solutions of the linearized original and extended Boussinesq equations led to the formulation of similarity indices for baseflow recession analysis, including scaling rules, to predict the moments of baseflow response. Validation of theoretical recession parameters on real-world streamflow data is complicated due to limited measurement accuracy, changing boundary conditions, and the strong coupling between the saturated aquifer with the overlying unsaturated zone. However, recent advances are shown to have mitigated several of these issues. The extended Boussinesq equation has been successfully applied to represent baseflow dynamics in catchment-scale hydrological models, and it is currently considered to represent lateral redistribution of groundwater in land surface schemes applied in global circulation models. From the review, it is clear that Wilfried Brutsaert and Jean-Yves Parlange stimulated a body of research that has led to several fundamental discoveries and practical applications with important contributions in hydrological modeling.

Recently, Troch et al. [10] introduced the hillslope-storage Boussinesq (hsB) equation to describe subsurface flow and saturation along complex hillslopes. They demonstrated that numerical solutions of the hsB equation account explicitly for plan shape of the hillslope, by introducing the hillslope width function, and for profile curvature through bedrock angle and the hillslope soil depth function (see also Paniconi et al. [7] and Hilberts et al. [4]). This paper presents an analytical solution of the linearized hsB equation applicable to hillslopes with constant bedrock slope and exponential width functions. Then, the first four moments of the impulse response function (travel time distribution) of such hillslopes are derived using the Laplace transform of the partial differential equation describing the groundwater flow. These moments are explicitly related to the hydraulic (porosity, conductivity) and geometric (bedrock slope, width function, and soil depth) parameters of a hillslope. The zeroth moment can obviously be interpreted as the total outflow volume, and the (normalized) first moment as the mean response time of the hillslope. The second, third, and fourth moments have interpretations in terms of the relative width, asymmetry, and peakedness, respectively, of the hydrograph. The paper examines the effect of the geometric properties of hillslopes on the travel time distribution in a dimensionless context.