Megascale thermodynamics in the presence of a conservative field: The watershed case

In a series of earlier papers the authors have proposed a unique approach for watershed modelling, which is based on developing watershed-scale balance equations for mass, momentum, energy and entropy by averaging the point-scale (microscale) equations over appropriate averaging regions or control volumes (megascale). The regions are referred to as Representative Elementary Watersheds (REWs), as they are considered to be invariant with respect to the spatial scale. Here, the REW-approach is generalized by developing balance equations and constitutive relationships for sub-REW units, referred to as Elements. Similar to an REW, Elements are divided into a series of zones to accommodate typical flow processes. The subdivision of an REW into Elements supports sub-REW-scale process representation. The proposed procedure yields exchange terms for mass, forces and thermal energy across phase and Element boundaries. These terms constitute unknowns and require a systematic closure. The closure is addressed within a thermodynamic approach, in which the Clausius–Duhem inequality formulated for a watershed serves as a mathematical and physical constraint. The present paper represents a clear extension of earlier work, as it includes non-isothermal processes in presence of the conservative gravitational field. The subdivision of an REW into Elements also provides means for including sub-REW variability due to landuse, geology or presence of infrastructure in the watershed. The paper also shows how an REW Element-scale unsaturated flow equation and non-linear reservoir equations for overland and channel flow can be consistently derived within the thermodynamic theory framework.