A TWO-STATE INTEGRAL-BALANCE MODEL FOR SOIL MOISTURE AND GROUNDWATER DYNAMICS IN COMPLEX TERRAIN

A dynamical model is devised for a hydrologic system where unsaturated and saturated storage serve as the principal control on rainfall-runoff and where complex topography, drainage area, and variable depth of moisture penetration describe the flow geometry. The model is formed by direct integration of the local conservation equation with respect to the partial volumes occupied by unsaturated and saturated moisture storage, respectively. This yields an “integral-balance” model in just two state variables. The relationship of the dynamical model to field data in complex terrain is found through a joint probability density for terrain features. This serves as a “volume” weighting function to construct conditional averages for the state variables and fluxes over a specified range of terrain features. The scale of averaging could range from hillslopes to river basins. Two examples of the joint probability of terrain features (altitude and aspect) are demonstrated for Valley, Ridge, and Appalachian Plateau digital elevation models. The strategy of a dynamical model formed by conditional averages of state variables with respect to terrain features is proposed as a way of simplifying the dynamics while preserving the natural spatial and temporal scales contributing to runoff response. The parametric form of the storage-flux or constitutive relationships for the proposed model is determined from numerical experiments in a simple hillslope flow geometry. The results show that a competitive relation exists between unsaturated and saturated storage except for the lowest precipitation rates. Saturation overland flow is proposed to be a storage-feedback relation. Solutions to the integral-balance model are presented in terms of the phase portrait, which represents all possible solution trajectories in state-space. The timing and magnitude of peaks in the runoff hydrograph from pulse-type input events demonstrate quick flow from near-stream saturated storage, saturation overland flow including rejected rainfall (storage-feedback), and late-time infiltration from upslope subsurface flow.