A unifying framework for watershed thermodynamics: Constitutive relationships

The balance equations for mass and momentum, averaged over the scale of a watershed entity, need to be supplemented with constitutive equations relating flow velocities, pressure potential differences, as well as mass and force exchanges within and across the boundaries of a watershed. In this paper, the procedure for the derivation of such constitutive relationships is described in detail. This procedure is based on the method pioneered by Coleman and Noll through exploitation of the second law of thermodynamics acting as a constraint-type relationship. The method is illustrated by its application to some common situations occurring in real world watersheds. Thermodynamically admissible and physically consistent constitutive relationships for mass exchange terms among the subregions constituting the watershed (subsurface zones, overland flow regions, channel) are proposed. These constitutive equations are subsequently combined with equations of mass balance for the subregions. In addition, constitutive relationships for forces exchanged amongst the subregions are also derived within the same thermodynamic framework. It is shown that, after linearisation of the latter constitutive relations in terms of the velocity, a watershed-scale Darcy's law governing flow in the unsaturated and saturated zones can be obtained. For the overland flow, a second order constitutive relationship with respect to velocity is proposed for the momentum exchange terms, leading to a watershed-scale Chezy formula. For the channel network REW-scale Saint-Venant equations are derived. Thus, within the framework of this approach new relationships governing exchange terms for mass and momentum are obtained and, moreover, some well-known experimental results are derived in a rigorous manner.