A unifying framework for watershed thermodynamics: constitutive relationships
Introduction
This work represents the sequel to a previous paper (Reggiani et al. [33]) concerned with the derivation of watershed-scale conservation equations for mass, momentum, energy and entropy. These equations have been derived by averaging the corresponding point scale balance equations over a well defined averaging region called the Representative Elementary Watershed (REW). The REW is a fundamental building block for hydrological analysis, with the watershed being discretised into an interconnected set of REWs, where the stream channel network acts as a skeleton or organising structure. The stream network associated with a watershed is a bifurcating, tree-like structure consisting of nodes inter-connected by channel reaches or links. Associated with each reach or link, there is a well-defined area of the land surface capturing the atmospheric precipitation and delivering it towards the channel reach. These areas uniquely identify the sub-watersheds which we define as REWs. As a result, the agglomeration of the REWs forming the entire watershed resembles the tree-like structure of the channel network on which the discretisation is based, as shown schematically in Fig. 1.
The volume making up a REW is delimited externally by a prismatic mantle, defined by the shape of the ridges circumscribing the sub-watershed. On top, the REW is delimited by the atmosphere, and at the bottom by either an impermeable substratum or an assumed limit depth. The stream reach associated with a given REW can be either a source stream, classified as a first order stream by Horton and Strahler 26, 38, or can interconnect two internal nodes of the network, in which case it is classified as a higher order stream.
The size of the REWs used for the discretisation of the watershed is determined by the spatial and temporal resolutions, which are sought for the representation of the watershed and its response, as well as by the resolution of available data sets. Change of resolution is equivalent to a change of stream network density, while still maintaining the bifurcating tree-like structure. Consequently, the way we have defined the REWs with respect to the stream network assures the invariance of the concept of REW with change of spatial scale.
The ensemble of REWs constituting the watershed communicate with each other by way of exchanges of mass, momentum and energy through the inlet and outlet sections of the associated channel reaches. In addition, they can also communicate laterally through exchanges of these thermodynamic properties across the mantle separating them (through the soils). The REW-scale conservation equations are formulated by averaging the balance laws over five subregions forming the REW, as depicted in Fig. 2. These subregions have been chosen on the strength of previous field evidence about different processes which operate within catchments, their flow geometries and time scales. The five subregions chosen in this work are denoted as follows: Unsaturated zone, Saturated zone, Concentrated overland flow, saturated Overland flow and main channel Reach. The unsaturated and saturated zones form the subsurface regions of the REW where the soil matrix coexists with water (and the gas phase in the case of the unsaturated zone). The concentrated overland flow subregion includes surface flow within rills, gullies and small channels, and the regions affected by Hortonian overland flow. It covers the unsaturated portion of the land surface within the REW. The saturated overland flow subregion comprises the seepage faces, where the water table intersects the land surface and make up the saturated portion of the REW land surface. For more in-depth explanations about the concept of REW the reader is referred to Reggiani et al. [33].
The REW-scale balance equations obtained by the averaging procedure represent the various REWs as spatially lumped units. Hence these equations form a set of coupled non-linear ordinary differential equations (ODE), in time only; the only spatial variability allowed is between REWs. Any spatial variability at the sub-REW-scale is averaged over the REW and can be represented in terms of effective parameterisations in the constitutive equations to be derived in this paper.
In the course of the averaging procedure a series of exchange terms for mass, momentum, energy and entropy among phases, subregions and REWs have been defined. These terms are the unknowns of the problem. A major difficulty is that the total number of unknowns exceeds the number of available equations. The deficit of equations with respect to unknowns requires that an appropriate closure scheme has to be proposed, which will lead to the derivation of constitutive relationships for the unknown exchange terms. In this paper we resolve the closure problem by exploiting the second law of thermodynamics (i.e., entropy inequality) as a constraint-type relationship. This allows us to obtain thermodynamically admissible and physically consistent constitutive equations for the exchange terms within the framework of a single procedure, applied uniformly and consistently across the REWs. This approach for the derivation of constitutive relationships is known in the literature as Coleman and Noll [5] method.
The second law of thermodynamics constitutes an inequality representing the total entropy production of a system. The inequality can be expressed in terms of the variables and exchange terms for mass, momentum and energy of the system and is subject to the condition of non-negativity. Furthermore, the entropy inequality is subject to a minimum principle, as explained, for example, by Prigogine [32]. An absolute minimum of entropy production is always seen to hold under thermodynamic equilibrium conditions. Consequently, under these circumstances, the entropy production is zero. In non-equilibrium situations the entropy inequality has to assume always positive values, because the second law of thermodynamics dictates that the entropy production of the system is never negative. This imposes precise constraints on the functional form of the constitutive parameterisations and reduces the degree of arbitrariness in their choice.
The Coleman and Noll method has been successfully applied by Hassanizadeh and Gray for deriving constitutive relationships in the area of multiphase flow 22, 18, for flow in geothermal reservoirs [17], for the theoretical derivation of the Fickian dispersion equation for multi-component saturated flow [21] and for flows in unsaturated porous media 23, 19.
Next to the thermodynamic admissibility, a further guideline for the constitutive parameterisations is the necessity of capturing the observed physical behaviour of the system, including field evidence. For example, it has been shown that overland and channel flows obey Chezy-type relationships, where the momentum exchange between water and soil is given by a second order function of the flow velocity. Similarly, according to Darcy's law, the flow resistivities for slow subsurface flow can be expressed as linear functions of the velocity. We will show that the proposed constitutive parameterisations will lead, under steady state conditions, to a REW-scale Darcy's law for the unsaturated and the saturated zones, and to a REW-scale Chezy formula for the overland flow. In the case of flow in the channel network an equivalent of the Saint–Venant equations for a bifurcating structure of reaches will be obtained.
The final outcome of this paper is a system of 19 non-linear coupled ordinary differential equations in as many unknowns for every REW. This set of equations needs to be solved simultaneously with the equation systems governing the flow in all the remaining REWs forming the watershed. A coupled (simultaneous) solution of the equation systems is necessary, since the flow field in one REW can influence the flow field in neighbouring REWs through up- and downstream backwater effects along the channel network, and through the regional groundwater flow crossing the REW boundaries. In developing the constitutive theory presented in this paper, we will make a number of simplifying assumptions to keep the problem manageable. Especially, the theory focusses on runoff processes at the expense of evapotranspiration, and hence the treatment of the latter is less than complete. Thermal effects, effects of vapour diffusion, vegetation effects and interactions with the atmospheric boundary layer will be neglected. These will be left for further research.
Section snippets
REW-scale balance laws
REW-scale conservation laws for mass, momentum, energy and entropy for the five subregions occupying the REW have been derived rigorously by Reggiani et al. [33] for a generic thermodynamic property ψ. In addition, the balance laws have been averaged in time, to accommodate different time scales associated with the various flow processes occurring within the watershed. The generic conservation law for the α-phase (water, soil, gas) within the i-subregion of an REW can be formulated as follows:
The global reference system
The conservation equation of momentum for the various subregions constitute vectorial equations, defined in terms of components with respect to an appropriate reference system. In order to be able to employ the equation for the evaluation of the velocity, we have to introduce a global reference system and effective directions of flow, along which the vectorial equations can be projected.
Saturated zone: In Ref. [33] we have assumed that the regional groundwater flow can be exchanged between
Constitutive theory
The system of equations in Section 2, derived for the description of thermodynamic processes in a REW, comprises in total 24 balance equations for the water, solid and gaseous phases in the unsaturated zone, the water and the solid phases in the saturated zone and the water phase in the two overland flow zones and in the channel. The total number of available equations is given by 8 mass balance equations, 8 (vectorial) momentum balance equations and 8 energy balance equations, respectively. In
Closure of the equations
The balance equations for mass, momentum, energy and entropy for all five subregions of the watershed have been rigorously derived by Reggiani et al. [33]. Thanks to Assumption I we can generally omit the subscripts α, which indicate different phases, as we are dealing with water in all subregions as the only mobile phase. The balance equations and relative exchange terms for mass and momentum, reported throughout the subsequent sections, refer, therefore, to the water phase only. Subsequently,
Parameterised balance equation
In the previous section we have proposed possible parameterisations of the mass and momentum exchange terms. Here we project the momentum balance equations along the reference system introduced in Section 3, and substitute the respective exchange terms for mass and momentum into the respective conservation equations. The mass balance equations have been derived by Reggiani et al. [33] and are written in the general form Eq. (2.2). For reason of simplicity of the final set of equations, we state
Discussion of the equation system
In the previous section we have obtained a system of 13 non-linear, coupled balance equations for mass and momentum for each subregion in the kth REW, which respect the jump conditions for mass and momentum across the inter-phase, inter-subregion and inter-REW boundaries. The 19 unknowns of the system are:The 13 balance equations in Section 5are supplemented with , , for the evaluation of the pressures. Based on the
Conclusions
This work is a sequel to a previous paper by Reggiani et al. [33] in which a systematic approach for the derivation of a physically-based theory of watershed thermodynamic responses is developed. In this approach, a watershed is divided into a number of subwatersheds, called Representative Elementary Watersheds (REWs). Each REW is, in turn subdivided into five subregions: unsaturated zone, saturated zone, concentrated overland flow zone, saturated overland flow zone, and a channel reach. A
Future perspectives
In concluding this paper, we feel the need to present an outline of where we believe the present work may be heading, and what we hope to achieve in the future. This seems appropriate, in order to give the readers a far-sighted, although somewhat biased and very much speculative vision of the long-term perspectives of our research, and to stimulate their interest and participation in some of the future goals.
In contrast to watershed hydrologists, fluid mechanicians have at their disposal well
Acknowledgements
P. Reggiani was supported by an Overseas Postgraduate Research Scholarship (OPRS) offered by the Department of Employment, Education and Training of Australia and by a University of Western Australia Postgraduate Award (UPA). This research was also supported by a fellowship offered by Delft University of Technology, which permitted P.R. to spend a six month period in The Netherlands. W.G. Gray was supported by the Gledden Senior Visiting Fellowship of the University of Western Australia while
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On leave from the Department of Civil Engineering and Geological Sciences, University of Notre Dame, Notre Dame, IN 46556, USA.