Cj. Duffy, A 2-STATE INTEGRAL-BALANCE MODEL FOR SOIL-MOISTURE AND GROUNDWATER DYNAMICS IN COMPLEX TERRAIN, Water resources research, 32(8), 1996, pp. 2421-2434
A dynamical model is devised for a hydrologic system where unsaturated
and saturated storage serve as the principal control on rainfall-runo
ff and where complex topography, drainage area, and variable depth of
moisture penetration describe the flow geometry, The model is formed b
y direct integration of the local conservation equation with respect t
o the partial volumes occupied by unsaturated and saturated moisture s
torage, respectively. This yields an ''integral-balance'' model in jus
t two state variables. The relationship of the dynamical model to fiel
d 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 o
ver a specified range of terrain features. The scale of averaging coul
d range from hillslopes to river basins. Two examples of the joint pro
bability of terrain features (altitude and aspect) are demonstrated fo
r 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 si
mplifying the dynamics while preserving the natural spatial and tempor
al scales contributing to runoff response. The parametric form of the
storage-flux or constitutive relationships for the proposed model is d
etermined from numerical experiments in a simple hillslope flow geomet
ry. The results show that a competitive relation exists between unsatu
rated and saturated storage except for the lowest precipitation rates.
Saturation overland flow is proposed to be a storage-feedback relatio
n. Solutions to the integral-balance model are presented in terms of t
he phase portrait, which represents all possible solution trajectories
in state-space. The timing and magnitude of peaks in the runoff hydro
graph from pulse-type input events demonstrate quick flow from near-st
ream saturated storage, saturation overland flow including rejected ra
infall (storage-feedback), and late-time infiltration from upslope sub
surface flow.