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 Boussinesq's 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 Darcy's 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.