Arno Hilberts

Analytical solutions to a hillslope-storage kinematic wave equation for subsurface flow

Abstract Hillslope response has traditionally been studied by means of the hydraulic groundwater theory. Subsurface flow from a one-dimensional hillslope with a sloping aquifer can be described by the Boussinesq equation [Mem. Acad. Sci. Inst. Fr. 23 (1) (1877) 252–260]. Analytical solutions to Boussinesqs equation are very useful to understand the dynamics of subsurface flow processes along a hillslope. In order to extend our understanding of hillslope functioning, however, simple models that nonetheless account for the three-dimensional soil mantle in which the flow processes take place are needed. This three-dimensional soil mantle can be described by its plan shape and by the profile curvatures of terrain and bedrock. This plan shape and profile curvature are dominant topographic controls on flow processes along hillslopes. Fan and Bras [Water Resour. Res. 34 (4) (1998) 921–927] proposed a method to map the three-dimensional soil mantle into a one-dimensional storage capacity function. Continuity and a kinematic form of Darcys law lead to quasi-linear wave equations for subsurface flow solvable with the method of characteristics. Adopting a power function of the form proposed by Stefano et al. [Water Resour. Res. 36 (2) (2000) 607–617] to describe the bedrock slope, we derive more general solutions to the hillslope-storage kinematic wave equation for subsurface flow, applicable to a wide range of complex hillslopes. Characteristic drainage response functions for nine distinct hillslope types are computed. These nine hillslope types are obtained by combining three plan curvatures (converging, uniform, diverging) with three bedrock profile curvatures (concave, straight, convex). We demonstrate that these nine hillslopes show quite different dynamic behaviour during free drainage and rainfall recharge events.

Low-dimensional modeling of hillslope subsurface flow: Relationship between rainfall, recharge, and unsaturated storage dynamics

[1] We present a coupling between the one-dimensional Richards equation for vertical unsaturated flow and the one-dimensional hillslope-storage Boussinesq equation (HSB) for lateral saturated flow along complex hillslopes. Here the capillary fringe is included in the flow domain as an integral part of the Boussinesq aquifer. The coupling allows quantitative investigation of the role of unsaturated storage in the relationship between rainfall and recharge. The coupled model (HSB coupled) is compared to the original HSB model (HSB original) and a three-dimensional Richards equation (RE) based model (taken to be the benchmark) on a set of seven synthetic hillslopes, ranging from convergent to divergent. Using HSB original, the water tables are overestimated and the outflow rates are generally underestimated, and there is no delay between rainfall and recharge. The coupled model, however, shows a remarkably good match with the RE model in terms of outflow rates, and the delay between rainfall and recharge is captured well. We also see a clear improvement in the match to the water tables, even though the values are still overestimated for some hillslope shapes, in particular the convergent slopes. We show that for the hillslope configurations and scenarios examined in this paper it is possible to reproduce hydrographs and water table dynamics with a good degree of accuracy using a low-dimensional hydrological model.

Comment on “Influence of capillarity on a simple harmonic oscillating water table: Sand column experiments and modeling” by Nick Cartwright et al.

[1] Cartwright et al. [2005] and, in a preceding paper, Nielsen and Perrochet [2000] conducted an investigation to assess the effects of an oscillating water table on the effective porosity. Both studies present results from a thorough laboratory experiment and numerical simulations. We wish to comment on some of the findings and conclusions of Cartwright et al. [2005] with regard to the effect of capillarity and the response at very high and very low oscillation frequencies. We also present some results from a simple numerical solution to a nonhysteretic one-dimensional Richards equation model and compare these to the original data and the modeling results presented by Cartwright et al. [2005].