A unifying framework for watershed thermodynamics: · PDF fileA unifying framework for watershed thermodynamics: balance equations for mass ... [email protected] ... forcing is of critical

A unifying framework for watershedthermodynamics: balance equations for mass,

momentum, energy and entropy, and the secondlaw of thermodynamics

Paolo Reggiania,*, Murugesu Sivapalana & S. Majid Hassanizadehb

aCentre for Water Research, Department of Environmental Engineering, The University of Western Australia, 6907 Nedlands, AustraliabDepartment of Water Management, Environmental and Sanitary Engineering, Faculty of Civil Engineering,

Delft University of Technology, P.O. Box 5048, 2600GA Delft, The Netherlands

(Received 20 July 1997; revised 13 April 1998; accepted 2 May 1998)

The basic aim of this paper is to formulate rigorous conservation equations for mass,momentum, energy and entropy for a watershed organized around the channel network.The approach adopted is based on the subdivision of the whole watershed into smallerdiscrete units, called representative elementary watersheds (REW), and the formulationof conservation equations for these REWs. The REW as a spatial domain is divided intofive different subregions: (1) unsaturated zone; (2) saturated zone; (3) concentratedoverland flow; (4) saturated overland flow; and (5) channel reach. These subregions alloccupy separate volumina. Within the REW, the subregions interact with each other,with the atmosphere on top and with the groundwater or impermeable strata at thebottom, and are characterized by typical flow time scales.

Thebalance equationsare derived for water, solid andair phases in the unsaturatedzone,waterandsolidphasesinthesaturatedzoneandonlythewaterphaseinthetwooverlandflowzones and the channel. In this way REW-scale balance equations, and respective exchangeterms for mass, momentum, energy and entropy between neighbouring subregions andphases,areobtained.Averagingof thebalanceequationsover timeallowstokeepthetheorygeneral such that the hydrologic system can be studied over a range of time scales. Finally,the entropy inequality for the entire watershed as an ensemble of subregions is derived asconstraint-type relationship for the development of constitutive relationships, which arenecessary for the closure of the problem. The exploitation of the second law and thederivation of constitutive equations for specific types of watersheds will be the subject of asubsequent paper.q 1998 Elsevier Science Limited. All rights reserved

Keywords:representative elementary watersheds, subregions, balance equations.

NOMENCLATUREA mantle surface with horizontal normal delimiting the

REW externallyb external supply of entropy, [L2/T3o]CA external boundary curve of the REWCr length of the channel, [L]e mass exchange per unit surface area, [M/TL2]E internal energy per unit mass, [L2/T2]f external supply term forwF entropy exchange per unit surface area projection, [M/

T3o]

g the gravity vector, [L/T2]G production term in the generic balance equationh external energy supply, [L2/T3]i general flux vector ofwj microscopic non-convective entropy flux [M/T3o]L rate of net production of entropy, [M/T3oL]mr volume per unit channel length, equivalent to the aver-

age cross-sectional area, [L3/L]M number of REWs making up the entire watershedn jA unit normal pointing from thej-subregion outward with

respect to the mantle surfacen j unit normal pointing from thej-subregion into the

atmosphere or into the ground

Advances in Water ResourcesVol. 22, No. 4, pp. 367–398, 1998q 1998 Elsevier Science Ltd

Printed in Great Britain. All rights reserved0309-1708/98/$ - see front matterPII: S 0 3 0 9 - 1 7 0 8 ( 9 8 ) 0 0 0 1 2 - 8

367

*Corresponding author. Fax: 0061 8 9380 1015; e-mail:[email protected]

n ji unit normal pointing from thej-subregion into thei-subregion

nab unit normal pointing from thea-phase into theb-phaseN global outward normal to the mantle surfaceNk number of REWs surrounding thekth REWq heat vector, [M/T3]Q energy exchange per unit surface area projection,

½M=T3ÿ

s the saturation function, [—]S the time-invariant surface area of the REW, [L2]t microscopic stress tensor, [M/T2L]t time, [T]T momentum exchange per unit surface area projection,

[M/T2L]v velocity vector of the bulk phases, [L/T]V the global reference volume, [L3]Vj volume occupied by the entirej-subregion, [L3]w velocity vector for phase and subregion boundaries,

½L=Tÿ

yj average thickness of thei-subregion along the vertical,[L]

Greek symbols

g the phase distribution functionD indicates an increment in timee j porosity of thej-subregion soil matrix,eja j-subregiona-phase volume fraction,

h the entropy per unit mass, [L2/T2o]v the temperaturey r the length of the main channel reachCr per unit

surface area projectionS, [1/L]r mass density, [M/L3]S projection of the total REW surface areaS onto

the horizontal plane, [L2]S j horizontal projection of the surface area covered

by the j-subregions, [L2]w a generic thermodynamic propertyq j The time-averaged surface area fraction

occupied by thej-subregion, [—]

Subscripts and superscripts

i superscript indicating a subregion, theatmosphere or the underlying deep soil

j superscript which indicates the varioussubregions within a REW

k subscript which indicates the various REWswithin the watershed

l index which indicates the various REWssurrounding thekth REW

top superscript for the atmosphere, delimiting thedomain of interest at the top

bot superscript for the the region delimiting thedomain of interest at the bottom

a,b indices which designate different phasesm, w, g designate the solid matrix, the water and the

gaseous phase, respectively

Special notation∑bÞa

summation over all phases except thea-phase∑

l

summation over allNk REWs surroundingthe kth REW

〈〉ja average operator defined for the u- and thes-subregiona-phases

〈〉j average operator defined for the o-, c- andther-subregions

¹ ja mass-weighted average operator defined for

the u- and the s-subregiona-phases

¹ j mass-weighted average operator defined forthe o-, c- and the r-subregions

,ja deviation from the mass-weighted average

within the u- and s-subregiona-phase,j deviation from the mass-weighted average

within the o-, c- and r-subregionja denotes a quantity within the u- and the

s-subregiona-phasej denotes a quantity within the o-, c- and the

r-subregionj

a property for the u- and the s-subregiona-phase defined on a per unit area basis

j

property for the o-, c- and the r-subregionsdefined on a per unit area basis

{} k, [] k, ()k, lk denotes a quantity or expression relative tothe kth REW

Particular combinations of super- and subscripts for thetermse, I, T, Q andF; u- and s-subregion:

jAa exchange fromj-subregiona-phase across

the mantlejAa, l exchange from thej-subregiona-phase

across thelth mantle segmentjAa,ext exchange from thej-subregiona-phase

across the ext. watershed boundaryj topa exchange betweena-phase and atmospherej bota exchange betweena-phase and underlying

stratajab intra-subregion exchange betweena-phase

andb-phasejia inter-subregion exchange betweenj-subre-

gion andi-subregion

c-, o- and r-subregions

jA exchange fromj-subregion across the mantlejAl exchange from thej-subregion across thelth

mantle segmentjAext exchange from the

i-subregion across theexternal watershed boundary

ji inter-subregion exchange betweenj-subre-gion andi-subregion

j top exchange betweenj-subregion and atmo-sphere

368 P. Reggianiet al.

1 INTRODUCTION

The study of the response of a watershed to atmosphericforcing is of critical importance to applied hydrology andremains a major challenge to hydrological research. Amongthe current generation of hydrological models we can dis-tinguish two major categories which are called, in short,physically-based and conceptual models. Freeze13 intro-duced the first of a generation of distributed physically-based watershed models founded on rigorous numericalsolution of partial differential equations (PDE) governingflow through porous media (Richards’ equation, Darcy’slaw), overland flow (kinematic wave equation) and channelflow (Saint–Venant equations). These equations also formthe basis of other distributed watershed models such as theSysteme Hydrologique Europe´en (SHE) model described byAbbott et al.1,2 Others to follow similar approaches wereBinley and Beven8 and Woolhiseret al.35

There has been considerable discussion regarding theadvantages and disadvantages of such distributed physi-cally-based models by Beven,5 Bathurst4 and O’Connelland Todini.31 While these models have the advantage thatthey explicitly consider conservation of both mass andmomentum (but expressed at the point or REV scale),their main shortcomings are the following: first, the numer-ical solution of the governing equations is a computationallyoverwhelming task, except for very small watersheds; themodels also require detailed information about soil proper-ties and geometry, which is not available for most real worldwatersheds. Even if the necessary input information wasavailable, detailed results provided by distributed modelsare not needed for most practical purposes. Secondly,since these distributed watershed models are usually overparameterized, infinite combinations of parameter valuescan yield the same result, leading to a large parameter esti-mation problem (see e.g. Beven6).

Parallel to these distributed models, a series of lumpedconceptual watershed models have been developed. Thesedo not take into account the detailed geometry of the systemand the small-scale variabilities, rather they consider thewatershed as an ensemble of interconnected conceptualstorages. Examples are the Nash cascade30 or the StanfordWatershed model29. The disadvantages of many of the cur-rent lumped conceptual models are listed as follows. First,they are based only on the mass balance and do not expli-citly consider any balance of forces or energy. Secondly, themodels usead hocparameterizationsin lieu of derived con-stitutive relationships for the various mass exchange terms.As a result, the parameters lack physical meaning.

At present, there does not exist an accepted general fra-mework for describing the response of a hydrologic systemwhich is applicable directly at the spatial scale of awatershed and takes into account explicitly balances ofmass, momentum and energy (i.e. without having to usepoint-scale equations) while serving as a guideline formodel development, field data collection and design appli-cations. The aim of the present work is to attempt to fill this

gap by formulating an approach which aims to combine theadvantages of the distributed and lumped approaches.

The procedure we present here has been motivated by theaveraging approach for multi-phase flow in porous media,pioneered by Hassanizadeh and Gray, and published in aseries of papers, e.g.17–22. the physics of porous mediaflow must consider fluid motions through an interconnectedsystem of soil pores, and the physical and chemical interac-tions between the fluid phase, the gaseous phase and thesolid phase representing the soil matrix. However, thedetails of the complex arrangement of soil grains and thegeometry of the pore spaces are unknown, and generallyunknowable. Also, predictions of flow through porousmedia are required, not at the scale of the pore (the micro-scale), but at scales much larger than the pore (the macro-scale). For these reasons, Hassanizadeh and Gray17–19

sought to develop a theory of porous media flow which isexpressed in terms of balance equations for mass, momen-tum, energy, and entropy at the macroscale. At this scale theporous medium is then treated as a continuum. The aver-aging procedure has been successfully applied by Hassani-zadeh and Gray to derive Darcy’s law for single-phaseflow19 and for two-phase flow,22 and the Fickian dispersionequation for multi-component saturated flow.20 The aver-aging approach has also been referred to as ‘hybrid mixturetheory’ by Achantaet al.3 in a work regarding flow in clayeymaterials.

In the present paper, the averaging approach will beemployed in order to derive watershed scale balance equa-tions for mass, momentum, energy and entropy. The wholewatershed is divided into smaller entities over which theconservation equations are averaged in space and time.The distributed description of the watershed is replaced byan ensemble of interconnected discrete points. The ensem-ble of points has subsequently to be assembled by imposingappropriate jump conditions for the transfer of mass,momentum, energy and entropy across the various bound-aries of the system. Furthermore, the large range of timescales typical for the various flows within a watershedrequires additional averaging of the equations in time.

The averaging approach presented here has never beenpursued before in watershed hydrology. However, recently,Duffy10 presented a hillslope model based on the globalmass balance equations, formulated for a two-state (unsatu-rated and saturated store) system. Constitutive relationshipswere postulated for the mass exchange between the stores,and with the surroundings. The parameter calibration of themodel was performed by using a numerical Richards’ equa-tion solver. In our approach constitutive equations will bederived with the aid of the second law of thermodynamics,explicitly taking into account the equations for conservationof momentum and energy.

As a concluding remark we emphasize that theories forflow and transport are usually derived in a separate mannerfor the saturated and the unsaturated zones, for the overlandand the channel flows by writing flow equations for thevarious zones. The equations are later hooked together

Unifying framework for watershed thermodynamics 369

during modelling. In the present approach we obtain balanceand constitutive equations for these zones within the frame-work of a single procedure by ensuring the compatibility ofall constitutive relationships for the entire watershed withthe second law of thermodynamics. We believe this willlead to more consistent models in the future.

2 OUTLINE OF THE UNIFYING FRAMEWORK

The purpose of the present work is to propose a generalframework which can be recast to address specific problemsamong the entire spectrum of hydrologic situations. In awider sense, a watershed constitutes an open thermody-namic system, where mass, momentum, energy and entropyare permanently exchanged with the atmosphere or sur-rounding regions. These exchanges are driven by massinput from the atmosphere during storms, by extraction ofmass towards the atmosphere during interstorm periods, andby transfer of mass towards adjacent regions. Atmosphericforcing (rainfall, solar radiation) and gravity (run-off) playfundamental roles in these processes. It is important for thetheoretical framework to be expandable, in order to handleother problems related to the hydrologic cycle. In this con-text one could think of terrestrial water and energy balances,transport of sediments and pollutants, land erosion, salinityproblems and land–atmosphere interaction at the watershedscale, relevant for the implementation of global climatemodels (GCM). Even though the present work is restrictedto the surface and subsurface zones and does not explicitlyconsider the presence of vegetation or the transport of sedi-ments or chemical species, future inclusion of these issues iscompatible with the developments pursued here.

As pointed out by Gupta and Waymire,16 the most strik-ing feature observed in a watershed is the interconnectedsystem of hillslopes organized around the river network.Hillslopes convert a fraction of the rainfall into run-offwhile the remainder becomes part of the soil moisture sto-rage. Run-off generated from hillslopes is transferred by thechannel network towards a common point, called thewatershed outlet. A part of the soil moisture held withinthe hillslopes percolates down to recharge the regionalgroundwater reservoir. The remaining part is removed asevapotranspiration (bare soil evaporation and plant tran-spiration), providing for a return of water vapour andlatent thermal energy back into the atmosphere and so con-tributing to the maintenance of the global hydrologic cycle.In a simplistic view one might envisage the watershed as alarge funnel which in turn is composed of a number ofsmaller funnels. Each one of these smaller funnels can bedecomposed again into even smaller ones and so forth. Inpursuing an averaging approach, these sub-watershed scalefunnels form natural averaging regions.

With these considerations in mind, in our work the wholewatershed is divided into a number of smaller sub-water-sheds which we refer to as representative elementary water-sheds (REWs). The averaging of the conservation equations

is carried out over the REW. Clearly, the system contains alarge amount of spatial variability at scales smaller than theREW. By choosing the REW as the fundamental unit ofdiscretization in space, we incorporate the effects of suchsmall-scale variabilities in an effective manner and contem-plate variability only between various REWs.

Another important issue to be considered in the context ofsuch a framework is the range of time scales characterizingthe hydrology of a watershed. It is clear that hydrologicalprocesses associated with the two fundamental componentsof a watershed, i.e. hillslopes and the channel network,typically operate over vastly different time scales. This isdue to the variety of media in which water movement takesplace. Even within a single hillslope there are different timescales associated with the various pathways the water takesto travel through the hillslope and exit the system. Theseinclude surface (overland flow) and subsurface pathways(unsaturated and saturated zone, regional groundwater sys-tem), and evapotranspiration. In Table 1 typical velocitiesassociated with the various flow pathways are listed. Theproblem of dealing with a large range of time scalesbecomes especially apparent when balance equations usedto describe the different hydrological processes (e.g. infil-tration, surface run-off, channel flow, evapotranspiration)need to be combined together. Moreover, for the samewatershed, one may be interested in short-term or long-term events. The time scale associated with the responseof a watershed to a single rainfall event is much smallerthan that of seasonal events (e.g. snowmelt, droughts) orlong-term water yield. This implies that in the first casesome processes occurring at very low velocities (e.g. filtra-tion through the unsaturated zone) become unimportantwith respect to the faster processes (e.g. surface run-off orflow in the channel network) which determine the behaviourof the system within the time frame of the event. In thesecond and third cases, fluctuations of the dynamic variablesdue to forcing events which are short with respect to thetime scale under consideration can be averaged out withoutobscuring the long-term variations.

Vis avis the wide range of time scales at which the systemcan operate, we propose to average the governing equationsover a time interval which is chosen according to the parti-cular application. The ‘rapid’ fluctuations of the system attime scales smaller than the averaging interval will not beaccounted for and are filtered out. This approach is widely

Table 1. Comparison of the flow pathways (based on data byDunne, 197811)

Flow type Temporal scale Velocity [m s¹1]

Groundwater flow Days–years # 10¹6

Hortonian overlandflow

Hours 10¹3 –10¹1

Subsurface flow Hours–days 10¹7–10¹4

Saturated overlandflow

Hours 10¹2–10¹1

Channel flow Hours–days 1–10


used in the study of turbulent flows, where time averages ofthe fluctuating variables (e.g. pressure, density, velocity) areperformed over a characteristic time interval. The averagingprocedure developed here results in a set of conservationequations of mass, momentum, energy and entropy for eachsubregion of the REW. The conservation equations willhave the following general form:

dw

dt¼

∑i

ewi þ Rþ G (1)

where w can be such properties as soil water saturation,water mass density, flow velocity, water energy, waterdepth etc.,ew

i is the exchange of mass, force, or heatamong various phases and subregions,R is a possible exter-nal supply, andG accounts for internal generation. Theforms of these general balance equations are of coursewell known. The difficulty lies in the determination ofrelationships between the exchange terms andw-quantities.The approach presented here provides a framework fordeveloping such relationships based on physical laws, aswill be shown in a sequel paper. Substitution of these rela-tionships in the balance laws will result in equations gov-erning various flow mechanisms. Relationships analogousto well-known formulas such as Darcy’s law or Chezy’sformula are expected to be obtained at the watershed scaleas results of our approach.

3 THE REPRESENTATIVE ELEMENTARYWATERSHED (REW)

The present section is dedicated to the definition of thefundamental discretization units, the REWs, for which con-servation equations will be derived. Since we want to studyhydrological processes in watersheds, we need to define theaveraging volume in such a way as to preserve an autono-mous, functional watershed unit. In this respect it is naturalto refer to the drainage network as the basic organizingstructure for subdividing the watershed into smaller entities.This can be achieved by first disaggregating the channelnetwork into single reaches which connect two internalnodes of the network (classified as higher-order streamsby the Horton-Strahler network ordering system27,33). Inthe case of source streams (first-order streams accordingto the Horton–Strahler system), they will have an isolatedendpoint at their upper end and merge with another streamat a node further downstream. One can associate with everyreach a certain patch of land-surface, which drains watertowards it. This patch of land-surface is delimited externallyby ridges and thus remains a separate sub-watershed initself. We call the sub-watershed a representative elemen-tary watershed (REW) for the following reasons:

1. The REW includes all the basic functional compo-nents of a watershed (channels, hillslopes) and con-stitutes, therefore, a single functional unit, which isrepresentative of other sub-entities of the entire

watershed due to its repetitive character.2. The REW is the smallest, and therefore the most ele-

mentary unit, into which we discretize the watershedfor a given scale of interest.

The REW can also be assumed as being self-similar to thelarger basin to which it is subordinated, in the sense that itreveals similar structural patterns independently of the scaleof observation. This assumed self-similarity implies thatcertain aspects of composition and structure are invariantwith respect to the change in spatial scale. This propertycan, for example, be recognized most clearly in the channelnetwork. With increasing magnification, new branches andtreelike structures can be recognized, which are reminiscentof the same geometrical patterns manifested at a largerscale. These fine network branches cover even single hill-slope faces in the form of rills and gullies and are, therefore,part of the whole drainage network. The intrinsic self-simi-larity of the network and the related scaling properties arethe subject of extensive, ongoing research. Ample discus-sion of the topic and respective references to related litera-ture can be found in a recent treatise by Rodriguez-Iturbeand Rinaldo.32

It is relevant in this context to note that our definition ofthe fundamental unit, the REW, contemplates the self-simi-lar structure of the system and preserves the geometricinvariance with respect to change of spatial scale. This sug-gests that, for a given watershed, the REW can be chosen,for example, to be the entire watershed, or any smaller sub-watershed. The smallest admissible size of the REW is onein which the subregions, to be defined later in Section 4, arestill identifiable. Furthermore, the size of the REW is one inwhich the subregions are also dependent on the spatial andtemporal resolution one wants to achieve and on the spatialand temporal detail of the available data sets.

Fig. 1(a)–(c) show examples of how a real worldwatershed (Sabino Canyon, Santa Catalina Mountains, SE-Arizona) can be discretized into different sized REWs. Eachtime, we identify a main channel reach (solid grey line)associated with the REW and a sub-REW-scale network(dashed lines). The REW boundaries are indicated with asolid black line. In the first case (Fig. 1(a)) the REW coin-cides with the whole watershed. There is only a single mainstream and the catchment boundaries overlap with the REWboundaries. In the second and third cases (Fig. 1(b) and (c))the watershed has been subdivided into a finite numberM ofsmaller REWs: We decide to label the REWs with an indexk, k ¼ 1,…,M. Some REWs have a first order channel asso-ciated with them and possess, therefore, only one outlet.Other REWs are associated with higher order streams. Asa result they have an inlet as well as an outlet. It is importantto note that the main channel reach in a given REW maybecome part of a sub-REW-scale network in a larger REW,associated with a coarser discretization of the watershed.

As mentioned earlier, every REW is lumped into asingle discrete point in the process of the averaging pro-posed here. The original watershed will be substituted by


Fig. 1. (a) A real world watershed constituting a single REW (Sabino Canyon, Santa Catalina Mountains, SE-Arizona), reproduced from36). (b) The watershed of (a) discretized into 5 REWs; and (c) the watershed of (a), discretized into 13 REWs.


an agglomeration ofM points. The points are mutually inter-connected by plugging the outlet sections of two upstreamREWs into the inlet of a downstream REW. In this fashion,the typical tree-like branching structure of the network ispreserved. Fig. 2 shows an example of how REWs areassembled in the case of the watershed in Fig. 1(c), whichhas been subdivided intoM ¼ 13 sub-entities.

In the absence of surface erosion, the area of land cover-ing the kth REW is considered as a time-invariant spatialdomain and is denoted byS. Fig. 3 shows a 3-D view of asmall watershed discretized into three REWs, two of themrelative to first order streams, and one relative to a secondorder stream. The surfaceS is circumscribed by a curvelabelledCA and is considered coincident with the naturalboundaries of the REW (i.e. ridges and divides). The rainfallreaching the ground within these boundaries is drainedtowards a common outlet located at the lowest point ofCA, or is transferred to a groundwater reservoir. EachREW communicates with neighbouring REWs through themain channel reach which crosses the boundary at the inletand the outlet sections, and through mutual exchange ofgroundwater laterally. Next, a prismatic reference, volumeV is associated with the REW, confined by a mantle surfaceA. The mantle is defined by the shape of the curveCA andhas an outward unit normalN, which is at every point hor-izontal. On top the volumeV is separated from the atmos-phere by the surfaceS. At the bottom it is delimited either bythe presence of impermeable strata or by some limit depth

reaching into the groundwater reservoir. The limit depthcan, for example, be chosen as a common datum for theensemble ofM REWs forming the watershed. ThekthREW has a number ofNk neighbouring REWs and canhave a part of its mantle surfaceA in common with theexternal boundary of the watershed. Therefore, the mantlesurfaceA can be subdivided into a series of segments:

A¼∑

l

Al þ Aext (2)

where Al is the mantle segment forming the commonboundary between thekth REW and its neighbouringREW l. The segmentAext is the part of mantle, which thekth REW has in common with the external watershedboundary. For example, REW 1 in Fig. 1(c) has a mantlemade up of four segments, one of which is in common withthe external watershed boundary,Nk ¼ 3 and l assumes thevalues 2, 3 and 4. We observe thatAext is non-existent forREWs which have no mantle segment in common with theexternal boundary (e.g. REWs 4, 5 and 7 in Fig. 1(c)). Theentire space outside the watershed is referred to asexternalworld.

4 THE REW-SUBREGIONS

In previous sections, we emphasized that a watershedincludes two basic components: the channel network andthe hillslopes. Whereas observation and measurement ofchannel flow is relatively straightforward, there are a seriesof flow mechanisms occurring in the subsurface zone whichare difficult to quantify. Subsurface flow together with otherprocesses on the land surface (e.g. infiltration, evapotran-spiration, depression storages, surface detention, overlandflow), forms the hydrological cycle of a hillslope. Horton(see Refs.25,28,26) was amongst the first to try to quantify thehillslope hydrological cycle by analysing these processes onan individual basis. Horton explained the production of sur-face run-off by assuming that a rapidly decreasing infiltra-tion capacity of the soil would lead to infiltration excess run-off (Hortonian overland flow) on the surface. This modelhas been improved through the observation by Hewlett andHibbert23 of saturation excess run-off (saturated overlandflow) along the groundwater seepage faces in the lowerparts of the hilIslopes. Hewlett and Hibbert’s findingshave been further underpinned by extensive field investiga-tions carried out by Dunne.11

The common description of hillslope hydrological pro-cesses (see e.g. Chorley9) assumes the presence of a satu-rated zone in the subsurface region, delimited by the watertable. It is part of a regional groundwater which can haveextensions reaching beyond the watershed boundaries whilecommunicating with the channel network. The soil betweenthe groundwater table and the land surface forms the unsa-turated zone, where the soil matrix coexists with the waterand the gaseous phase (air–vapour mixture). The subsurface

Fig. 2. Hierarchical arrangement of the 13 REWs from Fig. 1(c).


zone can be dissected by macropores and fractures, causedby discontinuities in the soil properties and by the presenceof plant roots. In particular situations it is possible toobserve the formation of a perched aquifer in the subsurfacezone in the form of localized lenses of saturated soil. Oftenthe water table of the perched aquifers are separated by lesspermeable strata from the groundwater reservoir. The watertable of the groundwater reservoir or of the perched aquifersreaches the soil surface in the lower regions, at the foot ofthe hillslope. It generates saturated zones surrounding thechannel. These zones, also called variable contributing areaby Beven and Kirkby,7 are subject to seasonal changes andcan vary even at the time scale of a storm event. Rainfallreaching the ground within these saturated areas cannotinfiltrate and joins the stream as saturated overland flow.In the remaining parts of the watershed, low infiltrationcapacities can generate Hortonian overland flow. In addi-tion, early concentration of surface water generates a net-work of rills, gullies and ephemeral streams on top of theunsaturated land surface, which merge with the saturatedoverland flow in the zones surrounding the main channel.

Motivated by these field observations, and in order todescribe various flow processes, the volumeV occupiedby the REW is divided into five subregions, for each ofwhich we will derive balance equations for mass momen-tum, energy and entropy. The subregions are identified,

based on different physical characteristics and on the var-ious time scales typical for the flow within each zone. Weemphasize that these subregions do not need to be mutuallyinterconnected, and can also be composed of disconnectedparts. The subregions will be identified with a prescriptj,wherej can assume the symbols u, s, o, r or c. The choice ofthese symbols is motivated by the names of the five subre-gions as described below:

The u-subregionis formed by the unsaturated zone. Itincludes those volumina of soil, water, and gas, confinedat the top by the land surface and at the bottom towardsthe saturated zone by the water table. A typical situationis depicted in Fig. 4. Interactions of the water phase withthe soil matrix and the gaseous phase at constant atmo-spheric pressure have also to be taken into consideration.

The s-subregion comprises the saturated zone andincludes the volumina of soil and water underlying the unsa-turated zone. In this case the water phase, in contrast to theunsaturated zone, coexists only with the solid phase. Thephysical upper boundary of this subregion is given by thewater table. In the near-channel regions the water tablereaches the soil surface, forming seepage faces, whichcause saturation excess run-off. The upper boundary ofthe saturated zone is, in these areas, coincident with theland surface. The bottom boundary of the saturated zoneis set either by a limit depth reaching into the groundwater

Fig. 3. Three-dimensional view of an ensemble of three REWs.


reservoir or by the presence of impermeable strata. In thecase of a bottom boundary formed by bedrock (see Fig. 4),the shape of the boundary is dictated by the natural topo-graphy of the substratum. In the case of a groundwaterreservoir, the boundary can be drawn as a horizontalplane. Situations where the topography of the bedrock incombination with a horizontal plane determine the shapeof the bottom boundary are also possible.

Theo-subregionis the volume of saturated overland flow,forming on the seepage faces and within the sub-REW-scalenetwork branches lying within the saturated portion ofthe land surface as depicted in Fig. 4. The extension ofthe o-subregion is clearly defined by the intersection curveof the water table with the land surface and by the contourcurves forming the edges of the main channel reach. Thesaturation excess flow is fed by concentrated overland flow,return flow from the saturated zone and direct precipitationonto the saturated areas.

The r-subregion is the volume occupied by the mainchannel reach for a given REW. The channel reach is fedfrom the adjacent saturated overland flow zone (o-subre-gion) by lateral inflow and through direct rainfall from theatmosphere. Across the bed surface, the channel canexchange mass by either recharging the saturated zone orby filtration from the saturated zone towards the channel.Both phenomena depend on the regional flow regime withinthe saturated zone.

Thec-subregionis the sub-REW-scale network of chan-nels, rills, gullies, ephemeral streams and areas of Hortonianoverland flow within the unsaturated portion of the landsurface, collectively called concentrated overland flow.They form a volume of water, flowing towards the main

channel and merging with the saturated overland flow.The introduction of this fifth zone is necessary in order topreserve the invariance of the REW with respect to spatialscaling and, therefore, to account for the self-similar natureof the system. The presence of the fifth subregion allows usto account in a lumped way for the entire sub-REW-scalenetwork of channels and gullies as well as Hortonian over-land flow.

The thermodynamic properties may be exchanged acrossinter-subregion boundaries (e.g. seepage areas, channel bed,channel edges) or inter-REW boundaries (mantle segmentsdefined in eqn (2)) with neighbouring REWS. Furthermore,within the unsaturated and saturated zones, the phasesexchange the properties across phase interfaces (i.e. thewater–soil, water–gas and solid–gas interfaces). All thesesurfaces are assumed to be without inherent thermodynamicproperties and, therefore, standard jump conditions applybetween phases, subregions and REWs.

5 AVERAGING NOTATION AND DEFINITIONS

The derivations given here are based on the global balancelaws written in terms of a generic thermodynamic propertyw at the microscale. For a continuum occupying an arbitraryvolumeV*, delimited by a boundary surfaceA*, the balanceequation forw is stated as follows12:

ddt

∫Vp

rwdV þ

∫Ap

n p ·[r(v ¹ w p )w ¹ i]dA

¹

∫Vp

rf dV ¼

∫Vp

GdV ð3Þ

Fig. 4. Detailed view of the five subregions forming a REW.


wheren* is the unit normal toA* pointing outward,v is thevelocity of the continuum,w* is the velocity ofA*, i is adiffusive flux andr is the the mass density of the conti-nuum. The quantitiesw, i, f and G have to be chosenaccording to the type of thermodynamic property consid-ered. For the equations of balance of mass, linear momen-tum, energy, and entropy, the appropriate microscopicproperties are listed in Table 2. The propertyE is themicroscopic internal energy per unit mass,t is the micro-scopic stress tensor,g is the gravity vector,q is the micro-scopic heat flux vector,h is the supply of internal energyfrom the outside world,h is the microscopic entropy perunit mass,j is the non-convective flux of entropy,b is theentropy supply from the external world andL is the entropyproduction within the continuum. We will refer to Table 2throughout the whole paper for all the five subregions.

The following notational conventions will be introducedhere for later use. The subregions are made up either bya single phase or constitute multi-phase systems with two(s-subregion) or three (u-subregion) coexisting phases. TheREW-scale thermodynamic property intrinsic to a phase isdenoted as where the superscriptj ¼ u, s, c, o, r denotes therespective subregion and the Greek lettera ¼ w, m, g indi-cates the respective phase.

A single phase is confined through boundary surfacestowards the outside world, the neighbouring subregionsand the phases, which are present in the same subregion.The boundary surface, delimiting thej-subregion towardsthe external world on the mantle surfaceA, is denoted withAjA the surfaces separating it from the atmosphere on top orfrom the underlying bedrock or the soil–groundwater sys-tem at the bottom areAj

top andAjbot, and the surface forming

the inter-subregion boundary between thei and thej-sub-region is indicated with the symbolAij where the order ofthe superscripts is unimportant. Phase interfaces, on theother hand, are indicated by the symbolSj

ab, with a,b ¼

w, m, g, where the superscript indicates the subregion andthe Greek letters designate the two phases, which meet at theinterface. Here also the order ofa andb is unimportant.

Velocities of phases are indicated with the symbolv.Velocities of boundary surfaces are denoted withw. Thusthe velocity ofAjA is denoted withw jA, and those ofAj

top Ajbot

andAij with wjtop, wj

bot andw ji, respectively. The velocity ofphase interfaces is indicated withwab. Also in this case theorder of the indices is unimportant.

Unit normal vectors to the boundary surfaces are indi-cated withnjA for AjA, n j for Aj

top and Ajbot, n ji for Aji and

nab for Sjab, where the sequence of the indices is significant

insofar as it indicates the direction in which the vector ispointing. The unit normal vectorn jA is always pointing out-ward with respect to the mantle surfaceA, n ji is pointingfrom the j-subregion into thei-subregion, andnab from thea-phase into theb-phase. The vectorn j points outward intothe atmosphere or into the underlying layers.

In the same fashion, the sequence of the indices in thesuperscripts and subscripts is relevant for the REW-scaleexchange terms, which will be defined in the respectiveappendices. So, for example, the termeji

a is the a-phasemass source term for the subregion indicated by the firstsuperscript (j-subregion) through transfer from the subre-gion indicated by the second superscript (i-subregion).The term ej

ab on the other hand, is the intra-subregionmass source term for thea-phase through mass releasedfrom the b-phase. The mass exchange across the mantlesurface for the u- and the s-subregions, for example, isdenoted withejA

a . As a consequence of eqn (2) the totalmass flux can be written as a sum of fluxes across the seg-ments forming the mantle:

ejAa ¼

∑l

ejAa, l þ ejA

a,ext (4)

where the summation extends over theNk surroundingREWs and ejA

a,ext is non-zero only if the REW has amantle segment in common with the external watershedboundary. All other REW-scale exchange terms actalso as source terms for the respective thermodynamicproperty.

In general, when we deal with thekth REW we omit theindex k to keep the notation simple. If the ensemble ofMREWs is considered, the terms relative to thekth REW areput between brackets, which will then carry a subscriptk,e.g. ()k, [] k, { } k. Another possibility to indicate that a term isrelative to thekth REW is given by a vertical bar carryingthe subscript, e.g.lk.

The various subregions are space filling and occupyvolumes denoted withVj. At the REW-scale, though, theyappear as two-dimensional or one-dimensional domains.Therefore, a surfaceSj is associated with the u-, s- and theo-subregions, whereas a total lengthCr of the channel reachcan be attributed to the r-subregion. The projections of thesurface areasSj onto the horizontal plane is denoted withS j

and the projection of the whole surface areaSj with S. Thesurface areasSj and their projectionsS j are allowed toexpand or contract in time. The same is valid for the lengthCr of the main channel reach. The following ratios are intro-duced here:

q j ¼1

2DtS

∫t þDt

t ¹DtSj(t)dt j ¼ u,s,c,o (5)

are the time-averaged surface area fractions relative to thehorizontal projections for the u-, s-, c- and the o-subregions.Next, average values are defined for the various propertiesinvolved in the balance equations. The averages are carried

Table 2. Summary of the properties in the conservationequation

Quantity w i f G

Mass 1 0 0 0LinearMomentum

v t g 0

Energy E þ 1/2v2 t·v þ q h þ g·v 0Entropy h j b L


out over space and a characteristic time interval of length2Dt. For the unsaturated zone, saturated zone, and overlandflow subregions, an average vertical depthyj [L], is definedby:

y j ¼1

2DtqjS

∫t þDt

t ¹Dt

∫Vj

dVdt; j ¼ u, s,o,c (6)

where Vj denotes the space occupied by thej-subregion.The average mass density [M/L3] of phasea of the j-sub-region is defined by:

〈r〉y ja ¼

1

2Dte jaq jS

∫t þ Dt

t ¹ Dt

∫Vj

a

rdVdt; j ¼ u,s,o, c (7)

whereVja andej

a denote the space occupied by thea-phaseof the j-subregion and its corresponding volume fraction,defined by

e ja ¼

12DtyjqjS

∫t þ Dt

t ¹ Dt

∫Vj

gjadVdt; j ¼ u,s (8)

It should be remembered that in the case of the overlandflow and channel subregions, only water phase exists sothate j

a ¼ 1. For this reason the average mass density carriesonly a superscript, i.e.〈r〉 j, j ¼ o, c. Next, we introduce theaverage porositye j for the unsaturated zone (u-subregion).It is given by the sum of the volume fractions occupied bythe water and the gas phases:

eu ¼ euw þ eu

g (9)

whereas for the saturated s-subregiones ¼ esw. In addition, a

volume saturation function is introduced for the unsaturatedzone:

sua ¼ eu

a=eu a ¼ w,g (10)

subject to the condition:

suw þ su

g ¼ 1 (11)

For the channel subregion, the average water mass density[M/L3] is defined by

〈r〉r ¼ 12DtmryrS

∫t þ Dt

t ¹ Dt

∫Vr

rdVdt (12)

where mr and y r are the cross-sectional area [L2] and anaverage length measure [L¹1] of the channel reach, respec-tively, given by:

mr ¼1

2DtyrS

∫t þ Dt

t ¹ Dt

∫Vr

dVdt (13)

and

yr ¼1

2DtS

∫t þ Dt

t ¹ DtCr(t)dt (14)

The length measurey r is the time averaged length of themain channel reach per unit REW surface area projectionand is defined as drainage density by Horton.24,27 For thegeneric propertyw, expressed on a per unit mass basis, theaverage is defined through the expression

w ja ¼

1

2Dte jayj 〈r〉 j

aq jS

∫t þDt

t ¹Dt

∫Vj

rwgjadVdt; j ¼ u, x,o,c

(15)

where, once again, in the case of the c- and o-subregionsonly the water phase is present. This implies thate j

a ¼ 1 andthat the average quantity is denoted only with a superscript,i.e. wj , j ¼ o, c. For the channel reach, the average of thepropertyw is given by:

wr ¼1

2Dt〈r〉rmryrS

∫t þ Dt

t ¹ Dt

∫Vr

rwdVdt (16)

Finally, we observe that the average of the microscopicproperty f, expressed on a per unit mass basis, is evaluatedfor the various subregions in analogy to eqns (15) and (16),whereas the average of the entropy productionL is obtainedin a way similar to eqns (7) and (12), respectively.

6 REW-SCALE BALANCE EQUATIONS

The formulation of global balance laws for mass, momen-tum, energy and entropy at the scale of the REW has beenpursued in detail in the Appendices A, B, C, D, and E. In thissection only the final results will be presented and the mean-ing of the various REW-scale terms in the equations will beexplained. The groups of the four basic balance equationsare different from subregion to subregion and will, there-fore, be treated in separate sections. We recall that the mass,energy and entropy equations are scalar equations, whereasthe momentum balance is a vectorial equation. Furthermore,the unsaturated zone (u-subregion) includes water, gas (air–vapour mixture) and soil matrix as constituent phases,whereas in the saturated zone the water phase coexistsonly with the soil matrix. These considerations require thederivation of separate balance equations for every singlephase. In the study of watersheds only the water phase iscrucial, for which the equations are reported here. We recallthat the various equations listed below refer to thekth REWin an ensemble ofM REWs.

6.1 Unsaturated zone (u-subregion)

6.1.1 Conservation of massThe water mass balance equation for the unsaturated zone isderived from eqn (A19). It is reasonable to assume the com-plete absence of phase change phenomena between the solidphase and the remaining phases, within the aquifer (i.e. noabsorption, no solid dissolution, i.e.eu

sw ¼ eusg¼ 0). The pos-

sibility for mass exchange between water and gaseous phasewithin the soil pores has to be accounted for in order todescribe soil water evaporation. The resultant balanceequation for the water phase yields:

ddt

(ruwyueusu

wqu) ¼∑

l

euAw, l þ euA

w, ext þ eusw þ euc

w þ euwg

(17)

where ruw is the average water density,yu is the average

vertical thickness of the unsaturated zone,eu is the averageporosity of the soil matrix,su

w is the water phase saturation


andqu is the horizontal fraction of watershed area coveredby the unsaturated zone. The mass exchange terms repre-sent, in order of appearance, the exchange towards theneighbouring REWs across the mantle segments, theexchange across the external watershed boundary (non-zero only for REWs which have one or more mantle seg-ments in common with the external watershed boundary),the recharge to or the capillary rise from the saturated zone,the infiltration from the areas affected by concentratedoverland flow (i.e rills, gullies or Hortonian overlandflow), and the water phase evaporation or condensationwithin the soil pores, respectively.

6.1.2 Conservation of momentumThe next equation is given by the balance of forces acting onthe water body within the unsaturated zone. As mentionedbefore, the equation is vectorial and is subsequently asso-ciated with a resultant direction. The momentum balance forthe water phase is derived from eqn (A18) and is given in thegeneral form by eqn (A20). Multiplication of the mass con-servation eqn (A19) by the macroscopic velocityvu

w andsubsequent subtraction from the momentum balance (A20)yields:

(ruwyueusu

wqu)ddt

vuw ¹ ru

wyueusuwgu

wqu ¼∑

l

TuAw, l þ TuA

w,ext

þ Tusw þ Tuc

w þ Tuwg þ Tu

wm ð18Þ

The terms on the l.h.s are the inertial term and the waterweight, respectively. The r.h.s. terms represent variousforces: the total pressure forces acting on the mantle seg-ments in common with neighbouring REWs and with theexternal watershed boundary, the forces exchanged with theatmosphere and the deep groundwater, the forces trans-mitted to the saturated zone across the water table, to theconcentrated overland flow across the land surface, and,finally, the resultant forces exchanged with the gas phaseand the soil matrix on the water–gas and water–solid inter-faces, respectively.

6.1.3 Conservation of thermal energyThe REW-scale water phase conservation of thermal energyfor the unsaturated zone is derived from the conservation oftotal energy (eqn (A32)) by subtracting the balance ofmechanical energy (eqn (A33)). The result is:

(ruwyueusu

wqu)ddt

Euw ¹ ru

wyueusuwhu

wwu ¼∑

l

QuAw, l þ QuA

w, ext

þ Qusw þ Quc

w þ Quwg þ Qu

wm ð19Þ

where the terms on the l.h.s. are the energy storage due tochange in internal energy and the external energy supply(i.e. solar radiation, geothermal energy sources). The termson the r.h.s. are REW-scale heat exchange terms across themantle segments, the exchanges with the saturated zone,the concentrated overland flow, the gaseous phase and thesoil matrix, respectively.

6.1.4 Balance of entropyThe balance of entropy is given once again by multiplyingthe equation of mass conservation (eqn (A19)) by the REW-scale entropy and subtracting it subsequently from eqn(A34). The operation leads to the entropy equation in thefollowing form:

(ruwyueusu

wqu)ddt

huw ¹ ru

wyueusuwqu ¼ Lu

wqu

þ∑

l

FuAw, l þ FuA

w,ext þ Fusw þ Fuc

w þ Fuwg þ Fu

wm ð20Þ

where the the terms on the l.h.s. represent the entropy sto-rage and external supply, whereas the first term on the r.h.s.accounts for the internal production of entropy due to gen-eration of heat by internal friction. The remaining REW-scale entropy exchange terms express the interaction withthe surrounding REWs, subregions and phases in the sameorder as in the previous equations.

6.2 Saturated zone (s-subregion)

6.2.1 Conservation of massThe balance of water for the saturated zone is derived fromthe generala-phase mass conservation (eqn (B14)) inAppendix B. The possibility of accounting for regionalgroundwater movement has to be included. This requiresthat the total mass exchangeesA

w between the saturatedzones of two neighbouring REWs across the mantlesurface A can be non-zero. There is no precipitation orsolid dissolution between the soil matrix and the waterassumed, i.e.es

ws ¼ 0. The resultant conservation of massis given by:

ddt

(rswysesqs) ¼

∑l

esAw, l þ esA

w,ext þ es botw þ esu

w þ esow þ esr

w

(21)

whereys is the average vertical thickness of the saturatedzone. The r.h.s. terms are given in order of sequence: thefirst term represents the mass exchange across the mantlesegments forming the boundaries with neighbouringREWs, the second term is the flux across the mantle seg-ments in common with the external watershed boundary(non-zero only for REWs which have part of the mantlein common with the external watershed boundary), thethird term is the percolation to deeper groundwater zones,the fourth term is the flux across the water table(i.e. recharge or capillary rise), the fifth term is the fluxtowards the saturated overland flow zone on the seepageface and the last term is the mass exchanged with thechannel reach across the bed surface (i.e. lateral channelinflow through seepage and groundwater recharge throughchannel losses).

6.2.2 Conservation of momentumThe balance of forces for the saturated zone water phase isderived from eqns (B13) and (B15), in complete analogy to


what has been pursued for the unsaturated zone:

(rswysesqs)

ddt

vsw ¹ rs

wgswysesqs ¼

∑l

TsAw, l þ TsA

w,ext

þ Ts botw þ Tsu

w þ Tsow þ Tsr

w þ Tsws ð22Þ

The l.h.s. terms represent once again inertial force andweight of the water, whereas the r.h.s. terms are the ensem-ble of REW-scale forces, acting on the groundwater body,i.e. the forces exerted on the various mantle segments, theforces exchanged with the deep groundwater at the bottom,with the overland flow sheet on the seepage face, with thechannel across the bed surface and with the soil matrix onthe water–solid interfaces, respectively.

6.2.3 Conservation of thermal energyThe conservation of thermal energy is obtained from (B20),after subtraction of the mechanical energy:

(rswysesqs)

ddt

Esw ¹ rs

whswysesqs ¼

∑l

QsAw, l þ QsA

w,ext

þ Qs botw þ Qsu

w þ Qsow þ Qsr

w þ Qsws ð23Þ

where the r.h.s. terms are the various heat exchanges of thewater within the saturated zone with the neighbouringREWs, the external world, the surrounding subregionsand the soil matrix.

6.2.4 Balance of entropyThis balance law is derived from eqn (B27) after subtractionof the mass balance multiplied by the REW-scale entropy.The result is:

(rswysesqs)

ddt

hsw ¹ rs

wbswysesqs ¼ Ls

wqs þ∑

l

FsAw, l

þ FsAw,ext þ Fs bot

w þ Fsuw þ Fso

w þ Fsrw þ Fs

ws ð24Þ

where the r.h.s. terms do not require further explanations.

6.3 Concentrated overland flow (c-subregion)

6.3.1 Conservation of massThe concentrated overland flow region includes flow ofwater in the sub-REW-scale channel network (e.g. rillsand gullies), ephemeral streams and Hortonian overlandflow. It is modelled as a sheet of water covering the unsa-turated land surface. It receives rainfall and communicateswith the saturated overland flow around the main channelreach. The mass balance is derived in Appendix C from thegeneral balance eqn (C9). The result is:

ddt

(rcycqc) ¼ ec topþ ecu þ eco (25)

where yc is the average vertical thickness of the flowregion. The exchange terms on the r.h.s. are the inputfrom the atmosphere (i.e. rainfall), the infiltration into theunsaturated zone and the total mass flux into the saturatedoverland flow region. If desirable, it is possible to represent

the concentrated overland flow in terms of a drainage den-sity and an average cross-sectional area, instead of a flowdepth and an area fraction. This would require only smallconceptual changes by casting the equations into a similarform as for the case of the channel reach (r-subregion).

6.3.2 Conservation of momentumThe total momentum of the concentrated overland flow isbalanced by the gravity and the remaining forces acting onit. This is expressed by the following equation, which isderived from eqn (C11) after subtraction of the massbalance eqn (C10) multiplied by the average velocity ofthe c-subregion:

(rcycqc)ddt

vc ¹ rcycgcqc ¼ Tc topþ Tcu þ Tco (26)

where the three REW-scale forces on the r.h.s. are origi-nated by viscous interaction with the atmosphere, throughinteraction with the unsaturated zone (i.e. pressure and dragforce) and through exchange of momentum along the zoneswhere the sub-REW-scale network merges with the satu-rated overland flow zone, respectively.

6.3.3 Conservation of thermal energyThe conservation of total energy is derived from the con-servation of total energy (eqn (C14)) in the usual fashionthrough subtraction of the mechanical energy balance. Theresult is:

(rcycqc)ddt

Ec ¹ rcychcqc ¼ Qc topþ Qcu þ Qco (27)

where the r.h.s. terms are the REW-scale heat exchangeterms between the concentrated overland flow water bodyand the surroundings.

6.3.4 Balance of entropyFinally, the balance of energy obtained from eqn (C19) is:

(rcycqc)ddt

hc ¹ rcycbcqc ¼ Lcqc þ Fc topþ Fcu þ Fco

(28)

6.4 Saturated overland flow (o-subregion)

6.4.1 Conservation of massThe saturated overland flow mass balance is derived inAppendix D from the generic REW-scale balance eqn(D8). The resulting equation is:

ddt

(royoqo) ¼ eo topþ eor þ eoc þ eos (29)

where yo is the vertical average thickness of the watersheet. The r.h.s. terms represent, in order of appearance,the mass exchange with the atmosphere (i.e. rainfall andevaporation), the overland flow into the channel along thechannel edges, the inflow from the concentrated overlandflow regions uphill and the recharge of the flow sheetthrough seepage from the underlying saturated zone.


6.4.2 Conservation of momentumThe balance of forces is obtained from eqn (D10):

(royoqo)ddt

vo ¹ royogoqo ¼ To topþ Tor þ Toc þ Tos

(30)

where the inertial term and the total weight are balanced bythe following ensemble of resultant forces on the r.h.s.: thetotal force exchanged with the atmosphere on top, with thewater in the channel along the channel edges (i.e. drag andpressure force), with the concentrated overland flow on thecontact zones and the total force exchanged with thesaturated zone (i.e. pressure and drag force transmitted tothe soil matrix and the water in the saturated zone),respectively.

6.4.3 Conservation of thermal energyThe conservation law of thermal energy is derived in astraightforward manner form eqn (D13):

(royoqo)ddt

Eo ¹ royohoqo ¼ Qo topþ Qor þ Qoc þ Qos

(31)

6.4.4 Balance of entropyFinally, the balance of entropy is the result of manipulationsof eqn (D18):

(royoqo)ddt

ho ¹ royoboqo ¼ Loqo þ Fo top

þ For þ Foc þ Fos ð32Þ

6.5 Channel reach (r-subregion)

6.5.1 Conservation of massThe REW-scale mass balance equation for the channelreach is derived in Appendix E:

ddt

(rrmryr) ¼∑

l

erAl þ erA

ext þ er top þ ers þ ero (33)

wheremr is the average cross-sectional area of the reachandy r is thedrainage densitydefined in eqn (14). The r.h.s.terms are the various mass fluxes in and out of the channel.The first term is the sum of inflow and outflow at the inletand outlet sections of the REW. In the case of sourcestreams there will be only an outflow section. In the caseof higher order streams, there will be two inflows (fromeach of the two reaches converging at the REW inlet) andone outflow. The second term is the mass exchange of thechannel reach across the external watershed boundary. Thisterm is non-zero only for the REW situated at the outlet,e.g. REW 13 in Fig. 1(c). The third term is the massexchange between the channel free surface and the atmo-sphere (i.e. rainfall and open water evaporation). The fourthterm is the mass exchange with the saturated zone acrossthe channel bed and the last term is the lateral inflow intothe channel from the overland flow region.

6.5.2 Conservation of momentumThe momentum balance is derived from eqn (E16):

(rrmryr)ddt

vr ¹ rrmrgryr ¼∑

l

TrAl þ Tr top þ Trs þ Tro

(34)

It expresses the balance of forces between the inertial force,the weight of the water stored in the channel and the forcesacting on the surroundings. These are, in order of appear-ance, the total force acting on the end sections of thechannel by interaction with the upstream and downstreamreaches, the force exchanged across the external watershedboundary (non-zero only for the REW situated at thewatershed outlet), the force exerted by the atmosphere onthe free surface (i.e. wind stress), the force due to inter-action with the channel bed (i.e. pressure and drag force)and the total drag force exerted by the channel water on theoverland flow sheet along the channel edges.

6.5.3 Conservation of thermal energyThe balance of thermal energy is derived from eqn (E20)

(rrmryr)ddt

Er

¹ rrmrhryr ¼∑

l

QrAl þ QrA

ext þ Qr top þ Qrs þ Qro ð35Þ

where the interpretation of the heat exchange terms isstraightforward.

6.5.4 Balance of entropyThis final equation is obtained by manipulation of eqn (E26):

(rrmryr)ddt

hr ¹ rrmrbryr ¼ Lryr þ∑

l

FrAl þ FrA

ext þ Fr top

þ Frs þ Fro ð36Þ

7 RESTRICTIONS ON THE EXCHANGE TERMSFOR THE THERMODYNAMIC PROPERTIES

The previously defined subregions include one or morephases. The flow in the channel network and in the overlandflow region is a one-phase flow, whereas for the subsurfaceflow regions the coexistence of two or three phases has to beconsidered. The thermodynamic properties are exchangedbetween the different phases within the same subregion,between different subregions and among REWs.

Phases are separated by phase interfaces, subregions byinter-subregion boundary surfaces (e.g. the channel bed sur-face or the saturated land surface). REWs are separated fromeach other by the mantle surfaceA. These boundaries areassumed to have no inherent thermodynamic properties, i.e.they are not able to store any of the properties or to sustainstress. Under these circumstances the conservation equa-tions for mass, momentum, energy and entropy become,along these curves and surfaces, standard jump conditionsfor the conservation laws, as initially derived by Eringen12


and further generalized by Hassanizadeh and Gray.17,18 Inthe present case the following assumptions are made:

(1) The solid matrix is inert, i.e. there is nophase change or absorption between the soilmatrix and the remaining phases and, therefore,eu

mw ¼ euwm ¼ eu

mg ¼ eugm ¼ es

mw ¼ esmw ¼ 0:

(2) There are no sediment transport phenomena con-sidered (i.e. no surface or channel erosion) and, hence,the mass exchange terms between the solid phase ofthe u- and s-subregions with the two overland flowregions and the channel are zero:eso

m ¼ esrm ¼ euc

m ¼ 0.There are no mass exchanges of gas or solid phaseacross the mantle:euA

g ¼ euAm ¼ esA

m ¼ 0:(3) The solid matrix is a rigid medium, i.e.vu

m ¼ vsm ¼ 0, and the gaseous phase has negligible

motion, i.e.vug ¼ 0.

In assembling the subregions to REWs and the REWs tothe entire watershed, it has to be noted that the water phasecan exchange momentum, energy and entropy with theremaining phases (soil matrix, gas) within the same subre-gion and that the c-, o- and r-subregions, comprising onlythe water phase, interact with the water and the solid phaseof the adjacent u- and s-subregions. Appropriate jump con-ditions between different phases, subregions and REWshave, therefore, to be imposed.

Assumptions 1 and 2 allow the introduction of thefollowing notational simplifications: the mass exchangeterms between subregions are non-zero only for thewater phase. Therefore, the symbolseus

w , esuw ,

eucw , eso

w , esrw, esA

w, l , euAw, l , euA

w, ext andesAw, ext are substituted by

eus, esu, euc, eso, esr, esAl , euA

l , euAext andesA

ext, respectively.Following Assumption 3, only the velocities of the waterphase are considered in the unsaturated and the saturated zones.

We replace the symbolsvuw andvs

w for the water withvu andvs,respectively. The jump conditions are summarized in threetables: Table 3 contains the inter-phase jump conditions formass, momentum, energy and entropy for the u- and the s-subregion within thekth REW, Table 4 contains the respectivejump conditions between the five subregions within a REWand Table 5 summarizes the jump conditions between thekthREW and itslth neighbouring REW (l ¼ 1,…,Nk). Thesejump conditions will be employed for the manipulation ofthe second law of thermodynamics, pursued in Section 8.

8 THE SECOND LAW OF THERMODYNAMICS

In order to complete the series of balance laws for a REW,the second law of thermodynamics has to be included. It willbe useful for the derivation of constitutive relationships. Thesecond law of thermodynamics states that the rate of entropyproduction has to be non-negative for the physical systemunder consideration. In the present study the physicalsystem of interest is the entire watershed, which is madeup by an agglomeration ofM REWs. Every REW is an ensem-ble of five subregions. As a consequence the second law ofthermodynamics has to be written for allM REWs together:

L ¼∑Mk¼ 1

∑a ¼ m,w,g

∫Vu

LguadV

!k

þ∑Mk¼ 1

∑a ¼ m,w

∫Vs

LgsadV

!k

þ∑Mk¼ 1

∫Vc

LdV

� �k

þ∑Mk¼ 1

∫Vo

LdV

� �kþ

∑Mk¼ 1

∫Vr

LdV

� �k $ 0

ð37Þ

Introduction of REW-scale quantities defined in Section 5

Table 3. Inter-phase jump conditions for momentum, energy and entropy for theu and s-subregion

Subregion Property Boundary Jump condition

u Mass Suwg eu

wg þ eugw ¼ 0

u Momentum Sumg Tu

mg þ Tugm ¼ 0

Suwg (eu

wg·vuw þ Tu

wg) þ (eugw·vu

g þ Tugw) ¼ 0

Suwm Tu

wm þ Tumw ¼ 0

s Sswm Ts

wm þ Tsmw ¼ 0

u Energy Sumg Qu

mg þ Qugm ¼ 0

Suwg { eu

wg[Euw þ (vu2

w )=2] þ Tuwg·vu

w þ Quwg} þ { eu

gw[Eug þ (vu2

g Þ=2] þ Tugw·vu

g þ Qugw ¼ 0

Suwm ðTu

wm·vu þ Quwm) þ Qu

mw ¼ 0

s Sswm ðTs

wm·vs þ Qswm) þ Qs

mw ¼ 0

u Entropy Sumg Fu

mg þ Fumg $ 0

Suwg (eu

wg þ FuwgÞþ (eu

wghug þ Fu

gw) $ 0

Suwm Fu

wm þ Fuwm $ 0

s Sswm Fs

wm þ Fsmw $ 0


leads to:

L ¼∑Mk¼ 1

∑a ¼ m, w, g

LuaquS

!k

þ∑Mk¼ 1

∑a ¼ m,w

LsaqsS

!k

þ∑Mk¼ 1

(LcqcS)k þ∑Mk¼ 1

(LoqoS)k þ∑Mk¼ 1

(LryrS)k $ 0

ð38Þ

This inequality can be rewritten after eliminatingLu

a, Lsa, Lc, Lo andLr with the use of eqns (A34), (B27),

(C19), (D18) and (E26) and employing the equations ofmass conservation. After exploiting the inter-phase jumpconditions of Table 3 and dividing by the surface areaprojectionS we obtain:

∑Mk¼ 1

( ∑a ¼ m, w, g

�(ru

ayueuaqu)

ddt

hua ¹ FuA

a ¹ ruabu

ayueuaqu

¹ Fusa ¹ Fuc

a ¹ Fuab

��k

þ∑Mk¼ 1

( ∑a ¼ m,w

�(rs

aysesaqs)

ddt

hsa ¹ FsA

a

¹ rsabs

aysesaqs ¹ Fs bot

a ¹ Fsua ¹ Fso

a ¹ Fsra ¹ Fs

ab

��k

þ∑Mk¼ 1

(rcycqc)ddt

hc ¹ rcycbcqc ¹ Fc top¹ Fcu ¹ Fco� �

k

þ∑Mk¼ 1

�(royoqo)

ddt

ho ¹ royoboqo ¹ Fo top¹ Foc

¹ For ¹ Fos�

k

þ∑Mk¼ 1

(rryryr)ddt

hr ¹ FrArrmrbryr ¹ Fr top ¹ Frs ¹ Fro� �

k

$ 0 ð39Þ

Eqn (39) can be expressed in terms of the entropies and

Table 4. Inter-subregion jump conditions for mass momentum and energy and entropy

Property Boundary Jump condition

Mass Aus eus þ esu ¼ 0

Auc euc þ ecu ¼ 0

Aso eso þ eos ¼ 0

Asc esr þ ers ¼ 0

Aco eco þ eoc ¼ 0

Aor eor þ ero ¼ 0

Momentum Aus Tusm þ Tsu

m ¼ 0

(eus·vu þ Tusw ) þ (esu·vu þ Tus

w ) þ (esu·vs þ Tsuw ) ¼ 0

Auc (euc·vu þ Tucw þ Tuc

m þ Tucg ) þ (ecu·vc þ Tcu) ¼ 0

Aso (eso·vs þ Tsow þ Tso

mÞþ (eos·vo þ Tos) ¼ 0

Asr (esr·vs þ Tsrw þ Tsr

mÞþ (ers·vr þ Trs) ¼ 0

Aco (eco·vc þ TcoÞþ (eoc·vo þ Toc) ¼ 0

Aor (e∨·vo þ T∨Þþ (ero·vr þ Tro) ¼ 0

Energy Aus Qusm þ Qsu

m ¼ 0

{ eus[Eu þ (vu2)=2] þ Tusw ·vu þ Qus

w } þ { esu[Es þ (vs2)=2] þ Tsuw ·vs þ Qsu

w } ¼ 0

Auc { euc[Eu þ (vu2)=2] þ Tucw ·vu þ Quc

w þ Qucm þ Quc

g } þ { ecu[Ec þ (vc2)=2] þ Tcu·vc þ Qcu} ¼ 0

Aso { eso[Es þ (vs2)=2] þ Tsow ·vs þ Qso

w þ Qsom} þ { eos[Eo þ (vo2)=2] þ Tos·vo þ Qos} ¼ 0

Asr { esr[Es þ (vs2)=2] þ Tsrw·vs þ Qsr

w þ Qsrm} þ { ers[Er þ (vr2)=2] þ Trs·vr þ Qrs} ¼ 0

Aco { eco[Ec þ (vc2)=2] þ Tco·vc þ Qco} þ { eoc[Eo þ (vo2)=2] þ Toc·vo þ Qoc} ¼ 0

Aor { eor[Eo þ (vo2)=2] þ Tor·vo þ Qor} þ { ero[Er þ (vr2)=2] þ Tro·vr þ Qro} ¼ 0

Entropy Aus Fusm þ Fsu

m $ 0

(eushu þ Fusw ) þ (esuhs þ Fsu

w ) $ 0

Auc (euchu þ Fucw Þ þ ðFuc

m þ Fucg ) þ (ecuhc þ Fcu) $ 0

Aso (esohs þ Fsow Þþ Fso

m þ (eosho þ Fos) $ 0

Asr (esrhs þ Fsrw Þþ Fsr

m þ (ershr þ Frs) $ 0

Aco (ecohc þ Fco) þ (ecoho þ Foc) $ 0

Aor (eorho þ For) þ (erohr þ Fro) $ 0


internal energies expressed on a per unit REW area basis

hja ¼ rj

aejayjqjhj

a j ¼ u,s (40)

hj ¼ rjyjqjhj j ¼ c,o, r (41)

Eja ¼ rj

aejayjqjEj

a j ¼ u,s (42)

Ej ¼ rjyjqjEj j ¼ c, o, r (43)

At this stage one more assumption is made:

(4) The temperatures of all phases within the u- and thes-subregion are equal to a common temperaturevu andvs, respectively. The temperatures of the samesubregions are equal among allM REWs making upthe watershed, i.e.vj lk ¼ v jl l; j ¼ u,s,c,o,r;k,l ¼ 1…M.

This assumption allows for further notational simplifica-tions: vu

m ¼ vuw ¼ vu

g ¼ vu andvsm ¼ vs

w ¼ vs:

The entropy inequality in the form of eqn (39) can now berestated by formulating the entropy in terms of the internalenergy and the entropy on a per unit area basis, by exploit-ing the balance equations of thermal energy and by employ-ing the jump conditions summarized in Tables 4 and 5. Aftersome algebraic manipulations one obtains:

L ¼ ¹1vu

∑Mk¼ 1

dEum

dt¹ vu dhu

m

dt

� �k

¹1vu

∑Mk¼ 1

dEuw

dt¹ vu dhu

w

dt

� �k

¹1vu

∑Mk¼ 1

dEug

dt¹ vu dhu

g

dt

!k

¹1vs

∑Mk¼ 1

dEsm

dt¹ vs dhs

m

dt

� �k

¹1vs

∑Mk¼ 1

dEsw

dt¹ vs dhs

w

dt

� �k

¹1vc

∑Mk¼ 1

dEc

dt¹ vc dhc

dt

� �k

¹1vo

∑Mk¼ 1

dEo

dt¹ vo dho

dt

� �k

¹1vr

∑Mk¼ 1

dEr

dt¹ vr dhr

dt

� �k

¹∑Mk¼ 1

∑a ¼ m,w,g

ruayueu

aqu bua ¹

hua

vu

� �" #k

¹∑Mk¼ 1

∑a ¼ m,w,g

FuAa,ext ¹

QuAa, ext

vu

!" #k

Table 5. Jump conditions between thekth REW and the neighbouring REWs across the mantle segments

Property Boundary Jump condition

Mass AuAl euA

l lk þ euAk ll ¼ 0

AsAl esA

l lk þ esAk ll ¼ 0

Momentum AuAl TuA

m, l lk þ TuAm,kll ¼ 0

TuAg, l lk þ TuA

g,kll ¼ 0

(euAl ·vu þ TuA

w, l )k þ (euAk ·vu þ TuA

w,k)l ¼ 0

AsAl TuA

m, l lk þ TsAm,kll ¼ 0

(esAl ·vs þ TsA

w, l )k þ (esAk ·vs þ TsA

w,k)l ¼ 0

Energy QuAm, l lk þ QuA

m,kll ¼ 0

QuAg, l lk þ QgA

w,kll ¼ 0

{ euAl [Eu þ (vu2)=2] þ TuA

w, l ·vu þ QuA

w, l } k þ

{ euAk [Eu þ (vu2)=2] þ TuA

w,k·vu þ QuAw,k} l ¼ 0

AsAl QsA

m, l lk þ QsAm,kll ¼ 0

{ esAl [Es þ (vs2)=2] þ TsA

w, l ·vs þ QsA

w, l } k þ

{ esAk [Es þ (vs2)=2] þ TsA

w,k·vs þ QsAw,k} l ¼ 0

Entropy AuAl FuA

m, l lk þ FuAm,kll $ 0

FuAg, l lk þ FuA

g,kll $ 0

(euAl ·hu þ FuA

w, l )k þ (euAk ·hu þ FuA

w,k)l $ 0

AsAl FsA

m, l lk þ FsAm,kll $ 0

(esAl ·hs þ FsA

w, l )k þ (esAk ·hs þ FsA

w, k)l $ 0


¹∑Mk¼ 1

∑a ¼ m,w

rsayses

aqs bsa ¹

hsa

vs

� �" #k

¹∑Mk¼ 1

∑a ¼ m,w

Fs bota ¹

Qs bota

vs

!" #k

¹∑Mk¼ 1

∑a ¼ m,w

FsAa,ext ¹

QsAa, ext

vs

!" #k

¹∑Mk¼ 1

rcycqc bc ¹hc

vc

� �� k¹

∑Mk¼ 1

Fc top¹Qc top

vc

� �k

¹∑Mk¼ 1

royoqo bo ¹ho

vo

� �� k¹

∑Mk¼ 1

Fo top¹Qo top

vo

� �k

¹∑Mk¼ 1

rrmryr br ¹hr

vr

� �� k¹

∑Mk¼ 1

Fr top ¹Qr top

vr

� �k

¹∑Mk¼ 1

FrAext ¹

QrAext

vr

� �k

¹1vu

∑Mk¼ 1

("Tus

w þ Tucw þ Tu

wm þ Tuwg þ TuA

w

þ12(euc þ eusþ eu

wg þ∑

l

euAl )vu,R

�·vu,R

�k

¹1vs

∑Mk¼ 1

��Tso

w þ Tsrw þ Tsu

w þ Tswm þ TsA

w

þ12(esr þ esoþ esuþ

∑l

esAl )vs,Rÿ·vs,Rgk

¹1vc

∑Mk¼ 1

Tco þ Tcu þ12(ecu þ eco)vc, R

� �·vc, R

� �k

¹1vo

∑Mk¼ 1

�Tor þ Toc þ Tosþ

12(eoc þ eor þ eos)vo,R

� �

·vo,R�

k¹

1vr

∑Mk¼ 1

3 Tro þ Trs þ TrA þ12(ero þ ers þ

∑l

erAl )vr,R

" #·vr, R

( )k

¹1vu

∑Mk¼ 1

∑l

[(Euw ¹ vuhu

w)k ¹ (Euw ¹ vuhu

w)l ]euAl lk

¹1vs

∑Mk¼ 1

∑l

[(Esw ¹ vshs

w)k ¹ (Esw ¹ vshs

w)l ]esAl lk

¹1vr

∑Mk¼ 1

∑l

[(Er ¹ vrhr)k ¹ (Er ¹ vrhr)l ]erAl lk

þ1vu

∑Mk¼ 1

[(Euw ¹ vuhu

w)euAext]k

þ1vs

∑Mk¼ 1

[(Esw ¹ vshs

w)esAext]k þ

1vc

∑Mk¼ 1

[(Ec ¹ vchc)ec top]k

þ1vo

∑Mk¼ 1

[(Eo ¹ voho)eo top]k

þ1vr

∑Mk¼ 1

[(Er ¹ vrhr)erAext]k þ

1vr

∑Mk¼ 1

[(Er ¹ vrhr)er top]k

¹vu,s

vuvs

∑Mk¼ 1

(Qusw þ eusEu

w))k ¹vo, r

vovr

∑Mk¼ 1

(Qor þ eorEo)k

¹vu,c

vuvc

∑Mk¼ 1

(Qucw þ Quc

m þ Qucg þ eucEu

w)k

¹vc, o

vcvo

∑Mk¼ 1

(Qco þ ecoEc)k ¹vs, o

vsvo

∑Mk¼ 1

(Qsow þ Qso

m þ esoEsw)k

¹vs, r

vsvr

∑Mk¼ 1

(Qsrw þ Qsr

m þ esrEsw)k $ 0 ð44Þ

where:

vj, Ra ¼ vj

a ¹ vR; j ¼ u,s a ¼ m,w,g (45)

and

vj, R ¼ vj ¹ vR; j ¼ c,o, r a ¼ m,w,g (46)

are the velocities of the various phases with respect to areference velocityvR whereas

vj, i ¼ vj ¹ vi ; j, i ¼ u,s,c,o, r (47)

is the temperature difference between thej- and thei-sub-region. In subsequent steps, constitutive relationships willbe assumed for the external entropy supplies and theentropy exchanges between the various phases and the sur-roundings (i.e. atmosphere, soil–groundwater system andthe external world) for all subregions and REWs. Theinequality (eqn (44)) can be exploited as a constraint onthe form of constitutive relationships. This procedure willbe presented in detail in a future paper and will be appliedto derive a REW-scale theory of water flow.

9 CONCLUSIONS

An averaging procedure has been developed and appliedhere to formulate a unifying framework for the study ofwatershed thermodynamics. The watershed is discretizedinto elementary entities, representative elementary water-sheds (REW) by preserving the basic structure of the chan-nel network. The REW is a new concept introduced here forthe first time. Each REW is in fact a sub-watershed. Its sizemay vary from that of the entire watershed to the smallestidentifiable sub-watershed; that depends on spatial and tem-poral resolution of the data available, the type of applica-tion, and the time scale of the hydrologic phenomenon to be


studied. Each REW is subdivided into five subregions. Thesubdivision is motivated by hydrological field evidence andidentifies flow regions based on geometry and hydrody-namic regime.

REW-scale conservation equations for mass, momentum,energy and entropy have been derived for the ensemble ofphases in the unsaturated and the saturated zones and for thewater phase in the sub-regions concerned with overlandflow and channel flow. The equations are expressed interms of variables at the scale of the REW. The interactionsof a given phase with the remaining phases within a sub-region, with the adjacent subregions, and with the neigh-bouring REWs are accounted for through exchange terms ofmass, momentum, energy and entropy at the same scale.Additional averaging of the balance equations in timeallows to filter out fluctuations of the dynamic variables attime scales smaller than the scale typical for the specificprocess under study. The equations, furthermore, dependonly on time and represent the REW as being lumped intoa single point. The spatial structure of the various subsurfaceflow zones, overland regions and the channel are repre-sented by average quantities such as surface area fractionsand drainage density which are watershed-scale parametersmeasurable in the field.

The system of equations has a redundant number of vari-ables for which constitutive relationships are necessary. Thesecond law of thermodynamics will be needed as a con-straint on any proposed set of constitutive equations. Thesecond law has been derived here by combining the ensem-ble of phases, subregions and REWs for the wholewatershed together. The exploitation of the second lawwill be the subject of a subsequent paper. The formulationof generic balance equations, of the type presented in thispaper, has never been attempted before in watershed hydrol-ogy. Also, the fact that equations of momentum and energybalance will be explicitly used in the derivation of consti-tutive relationships is new. The procedure developed here isinvariant with respect to spatial scales and is flexible for thestudy of hydrological processes evolving over different tem-poral scales.

ACKNOWLEDGEMENTS

We are very grateful to Professor W. G. Gray for suggest-ing this approach in the first place and for his constructivecomments on early versions of the manuscript. We wish alsoto thank J. D. Snell for fruitful discussions and contributionsduring the early phase of this work. P. Reggiani was sup-ported by an Overseas Postgraduate Research Scholarship(OPRS) offered by the Department of Employment, Educa-tion and Training of Australia and by a University of Wes-tern Australia Postgraduate Award (UPA). This researchwas also supported by a travel award from the DistinguishedVisitors Fund of UWA to S. M. Hassanizadeh. Centre forWater Research Reference no. ED 1172 PR.

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APPENDIX A. CONSERVATION EQUATIONS FORTHE UNSATURATED ZONE (U-SUBREGION)

In the u-subregion the flow is delimited on top by the landsurface. It communicates with the saturated s-subregionacross the water table, as shown in Fig. 4, and is boundedlaterally by the mantle with a horizontal normal. The u-subregion exchanges the thermodynamic propertyw

across the water table with the saturated zone (s-subregion)and interacts laterally with the neighbouring REWs and theexternal world through the mantle. At the top it communi-cates with the rills, gullies and Hortonian overland flowareas forming the c-subregion. In the present case theglobal balance laws, written in terms of microscopic quan-tities, are applied directly to the whole u-subregion. Thesystem is here considered in the most general sense as athree-phase system due to the coexistence of a liquidphase w (water), a solid phase m (soil matrix) and a gaseousphase g (air–vapour mixture within the soil pores).

The balance law will be stated for all three phases, herelabelled asa-phases witha ¼ w, m, g. The integration overa phase is performed by making use of phase distributionfunctionsgj

a, for j ¼ u, s, as proposed by Gray and Lee.14

These functions are defined such that they assume the valueof 1 within the g-subregiona-phase and are 0 elsewhere.This approach allows the transfer of the integration limitsinto the integrand and, therefore, to integrate over less com-plex geometries. The use of a phase distribution function isnot needed for the c-, o- and r-subregions because only thewater phase is present there.

With reference to Fig. 4, the balance law for ana-phasethermodynamic property within the u-subregion is writtenas follows:

ddt

∫Vu

rwguadV þ

∫AuA

nuA·[r(v ¹ wuA)w ¹ i]guadA

þ

∫Auc

nuc·[r(v ¹ wuc)w ¹ i]guadA

þ

∫Aus

nus·[r(v ¹ wus)w ¹ i]guadA

þ∑bÞa

∫Su

ab

nab·[r(v¹wab)w¹ i]guadSþ

∫Vrfgu

adV

¼

∫Vu

GguadV; a ¼ m,w,g ðA1Þ

where the inter-subregion boundary surfacesAuA and Auj,the phase interfacesSu

ab, the unit normal vectorsnuA andnuj

and the velocitiesvua, wuA, and wuj and wab have been

defined in Section 5. The inter-subregion boundaryAus isformed by the water table andAuc by the part of land sur-face affected by concentrated overland flow. The summation∑

bÞa extends over all phases different from thea-phase.Eqn (A1) is successively averaged in time by integratingeach term separately over the interval (t ¹ Dt, t þ Dt) anddividing by 2Dt. According to a well-known theorem (seeWhitaker34, pp. 192–193), the order of time integration andtime differentiation may be changed so that the first term ineqn (A1) becomes:

12Dt

ddt

∫t þ Dt

t ¹ Dt

∫Vu

rwgsadVdt

¼1

2Dtddt

∫t þDt

t ¹Dt

∫Vu

rwguadVdt ðA2Þ

Now, applying the definition of average quantities Section(5)–(8) and (15), introduced in Section 5, one obtains:

12Dt

ddt

∫t þ Dt

t ¹ Dt

∫Vu

rwguadVdt ¼

ddt

(euayuqu , r .u

a wuaS)

(A3)

Next, a series of REW-scale exchange terms can be definedfor the u-subregion:

euAa ¼

12DtS

∫t þDt

t ¹Dt

∫AuA

nuA·[r(wuA ¹ v)]guadAdt;

a ¼ m, w,g ðA4Þ


is the REW-scale mass exchange of the u-subregiona-phase across the mantleA, while

I uAa ¼

12DtS

∫t þ Dt

t ¹ Dt

∫AuA

nuA·[i ¹ r(v ¹ wuA)wua]gu

adAdt

a ¼ m,w,g ðA5Þ

is the non-convective exchange of the propertyw of thea-phase across A. The quantitywu

a is the deviation ofw fromits space and time average value:

wua ¼ w ¹ wu

a (A6)

We observe that the exchange terms defined through eqns(A4) and (A5) can be rewritten as a sum of exchangesacross the various segments forming the mantle, as shownin eqn (2). As a result we can state the following equalities:

euAa ¼

∑l

euAa, l þ euA

a,ext (A7)

IuAa ¼

∑l

IuAa, l þ IuA

a, ext (A8)

where the sum extends over theNk mantle segments whichthe REW has in common with neighbouring REWS. Thesecond term on the r.h.s. is non-zero only for those REWs,which have one or more mantle segments in common withthe external boundary of the watershed. It is implicit for allthe other balance equations of the unsaturated zone that theexchanges across the mantle can be separated into the com-ponents relative to each segment. Following on, the quan-tities defined as:

euja ¼

12DtS

∫t þ Dt

t ¹ Dt

∫Auj

nuj ·[r(wuj ¹ v)]guadAdt;

j ¼ s, c a ¼ m,w,g ðA9Þ

are the mass exchange terms of the u-subregiona-phasewith the saturated zone across the water tableAus, and withthe concentrated overland flow across the areas of landsurfaceAuc, whereas

Iuja ¼

12DtS

∫t þ Dt

t ¹ Dt

∫Aus

nuj ·[i ¹ r(v ¹ wuj)wua]gu

adAdt

j ¼ s,c a ¼ m,w,g ðA10Þ

are the non-convective interactions between the u-subre-gion a-phase, the underlying saturated zone and the con-centrated overland flow, respectively. In a similar fashion

euab ¼

12DtS

∫t þ Dt

t ¹ Dt

∫Su

ab

nab·[r(wab ¹ v)]guadSdt;

a,b¼ m,w,g ðA11Þ

is the mass exchange term between thea-phase and theremaining phases. The term defined as:

Iujab ¼

12DtS

∫t þ Dt

t ¹ Dt

∫Sus

ab

nab·[i ¹ r(v ¹ wab)wua]gu

adSdt;

a,b¼ m,w,g ðA12Þ

is the non-convective exchange of propertyw between thea-phase and the other phases, accounting for the inter-phase exchanges of momentum, energy and entropy.Substitution of the exchange terms into the previouslytime-averaged eqn (A1) and use of the average quantitieseqns (5)–(8) and (15), yields the generic balance law forthe u-subregiona-phase:

ddt

(〈r〉uayueuawu

aquS) ¹ (euAa wu

a þ IuAa )S ¹ 〈r〉uayueu

a f uaquS

¼ 〈G〉uaquS þ (eusa wu

a þ Iusa )S þ (euc

a wua þ Iuc

a )S

þ∑bÞa

(euabw

ua þ Iu

ab)S; a ¼ m, w, g ðA13Þ

In the interest of brevity the averaging symbols are omittedunless otherwise confusion arises. Thus, the followingREW-scale quantities are defined:

〈r〉ua ¼ rua (A14)

wua ¼ wu

a (A15)

f ua ¼ f u

a (A16)

〈G〉ua ¼ Gua (A17)

eqn (A14) can be recast after division by the total projectedsurface areao in the form:

ddt

(ruawu

ayueuaqu) ¹ (euA

a wua þ IuA

a ) ¹ ruaf u

ayueuaqu

¼ Guaqu þ (eus

a wua þ Ius

a ) þ (euca wu

a þ Iuca )

þ∑bÞa

(euabw

ua þ Iu

ab); a ¼ m,w,g

For the water, the general conservation equation can berewritten in terms of the water saturationsu

w and the aver-age porosityeu, defined by eqn (9):

ddt

(ruawu

wyusuweuqu) ¹ (euA

w wuw þ IuA

w ) ¹ ruaf u

wyusuweuqu ¼

(A18)

Guwqu þ (eus

w wuw þ Ius

w ) þ (eucw wu

w þ Iucw )

þ∑

b¼ s,g(eu

wbwuw þ Iu

wb)

In the subsequent paragraphs thea-phase conservationequations for mass, momentum, energy and entropy willbe derived for the u-subregion.

Appendix A.1. Conservation of mass

For thea-phase mass conservation within the u-subregionthe microscale properties have to be defined according toTable 2 such thatw ¼ 1, i ¼ 0, f ¼ 0 andG ¼ 0. Subse-quently, eqn (A18) is reduced to the general form of themass balance for the u-subregiona-phase:

ddt

(ruayueu

aqu) ¼ euAa þ eus

a þ euca þ eu

ab (A19)


The terms on the r.h.s. account for the mass exchange of thea-phase with the neighbouring REWs and theexternal world across the mantle, with the adjacent s- andc-subregions, and with the remaining phases within the u-subregion.

Appendix A.2. Conservation of momentum

The REW-scalea-phase equation for conservation ofmomentum is derived by selecting the microscale propertiesfrom Table 2 such thatw ¼ v, i ¼ t, f ¼ g and G ¼ 0.Substitution into eqn (A18) yields:

ddt

(ruavu

ayueuaqu) ¹ (euA

a vua þ TuA

a ) ¹ ruayueu

aguaqu

¼ (eusa vu

a þ Tusa ) þ (euc

a vua þ Tuc

a ) þ∑bÞa

(euabvu

a þ Tuab)

ðA20Þ

where:

TuAa ¼

12DtS

∫t þ Dt

t ¹ Dt

∫AuA

nuA·[t ¹ r(v¹ wuA)vua]gu

adAdt

(A21)

is the REW-scale momentum exchange of the u-subregiona-phase with the neighbouring REWs and the externalworld across the mantleA and

Tuja ¼

12DtS

∫t þ Dt

t ¹ Dt

∫Auj

nuj ·[t ¹ r(v¹ wuj)vua]gu

adAdt;

j ¼ s,c ðA22Þ

are the momentum exchange terms with the s-subregionacrossAus and with the c-subregion throughAuc, respec-tively. Furthermore,

Tuab ¼

12DtS

∫t þ Dt

t ¹ Dt

∫Su

ab

nab·[t ¹ r(v ¹ wab)vua]gu

adSdt;

a,b¼ m,w,g ðA23Þ

is the momentum exchange between thea-phase and theremaining phases filling the u-subregion. Multiplication ofthe mass conservation eqn (A19) by the average velocityvu

a

and subsequent subtraction from eqn (A20) gives themomentum balance in the following form:

(ruayueu

aqu)ddt

vua ¹ TuA

a ¹ ruayueu

aguaqu ¼ Tus

a þ Tuca

þ∑bÞa

Tuab ðA24Þ

Appendix A.3. Conservation of energy

The REW-scalea-phase equation for conservation ofenergy for the u-subregion is derived by defining the micro-scale properties after Table 2, such thatw ¼ E þ v2/2, i ¼

(t·v þ q), f ¼ (g·v þ h) andG ¼ 0. Substitution of these

quantities into eqn (A18) leads to:ddt

{ruayueu

a[Eua þ (vu

a)2=2]qu} ¹ { euAa [Eu

a þ (vua)2=2]

þ TuAa ·vu

a þ QuAa } ¹ ru

ayueua(gu

a·vua þ hu

a)qu

¼ { eusa [Eu

a þ (vua)2=2] þ Tus

a ·vua þ Qus

a }

þ { euca [Eu

a þ (vua)2=2] þ Tuc

a ·vua þ Quc

a }

þ∑bÞa

{ euab[Eu

a þ (vua)2=2] þ Tu

ab·vua þ Qu

ab} ðA25Þ

where:

Eua ¼ Eu

a þ (vua)2u

=2 (A26)

hua ¼ hu

a þ gua·vu

au (A27)

are the REW-scale internal energy and the total heat supplyfrom the external world, respectively. The r.h.s. terms ofthe last two equations are composed of the average ofmicroscopic values plus a term attributable to sub-REW-scale fluctuations. The definitions ofvu

a and gua are given

similarly to eqn (A6). In the case of eqn (A26) the secondterm on the r.h.s. is given by averaged sub-REW scaledeviations of kinetic energy and for eqn (A27) by devia-tions of velocity and gravity. Further, note the followingdefinitions:

QuAa ¼

12DtS

∫t þ Dt

t ¹ Dt

∫AuA

nuA·{ q þ t·vua ¹ r(v ¹ wuA)

3 [Eua þ (vu

a)2=2]}guadAdt ðA28Þ

is the energy exchange of the u-subregiona-phase acrossthe mantle. Next, the terms defined as

Quja ¼

12DtS

∫t þ Dt

t ¹ Dt

∫Auj

nuj ·{ q þ t·vua ¹ r(v ¹ wuj)

3 [Eua þ (vu

a)2=2]}guadAdt; j ¼ s, c ðA29Þ

are the energy exchanges with the s-subregion and the c-subregion, respectively, while

Quab ¼

12DtS

∫t þ Dt

t ¹ Dt

∫Suj

ab

nab·{ q þ t·vua ¹ r(v ¹ wab)

3 [Eua þ (vu

a)2=2]}guadSdt; a,b¼ m,w,g ðA30Þ

is the energy exchange between thea-phase and theremaining phases. The quantity

Eua ¼ Ea þ (Eu

a þ (vua)2u

Þ=2¼ Ea ¹ Eua (A31)

is the deviation of thea-phase internal energy from itsaverage value. Subtraction of the equation for conservationof mass (eqn (A19)), multiplied byEu

a þ (vua)2=2, from eqn

(A25) yields:

(ruayueu

aqu)ddt

[Eua þ (vu

a)2=2] ¹ (TuAa ·vu

a þ QuAa )

¹ ruayueu

a(gua·vu

a þ hua)qu ¼ (Tus

a ·vua þ Qus

a )

þ (Tuca ·vu

a þ Quca ) þ

∑bÞa

(Tuab·vu

a þ Quab) ðA32Þ


The mechanical energy balance equation is obtained byforming the inner product of the velocityvu

a with themomentum balance eqn (A24):

(ruayueu

aqu)ddt

½(vua)2=2] ¹ TuA

a ·vua ¹ ru

ayueua(gu

a·vua)qu

¼ Tusa ·vu

a þ Tuca ·vu

a þ∑bÞa

Tuab·vu

a ðA33Þ

Appendix A.4. Balance of entropy

The balance of entropy for the u-subregiona-phase isobtained by defining the microscale properties followingTable 2 asw ¼ h, i ¼ j , f ¼ b and G ¼ L. The resultingequation is:

ddt

(ruayueu

aqu) ¹ FuAa ¹ ru

ayueuabu

aqu ¼ Luaqu

þ (eusa hu

a þ Fusa ) þ (euc

a hua þ Fuc

a Þ þ∑bÞa

Fuab ðA34Þ

where:

hua ¼ hu

a (A35)

bua ¼ bu

a (A36)

Lua ¼ 〈L〉ua (A37)

are the average entropy and the REW-scale terms ofentropy supply and internal generation of entropy, respec-tively, while

FuAa ¼

12DtS

∫t þ Dt

t ¹ Dt

∫AuA

nuA·[j ¹ r(v ¹ wuA)hua]gu

adAdt

(A38)

is the entropy exchange across the mantleA, and

Fuja ¼

12DtS

∫t þ Dt

t ¹ Dt

∫Auj

nuj ·[j ¹ r(v ¹ wuj)hua]gu

adAdt

j ¼ s,c ðA39Þ

are the entropy fluxes into the s- and c-subregions, respec-tively. Finally,

Fuab ¼

12DtS

∫t þ Dt

t ¹ Dt

∫Su

ab

nab·[j ¹ r(v ¹ wab)hua]gu

adSdt;

a,b¼ m,w,g ðA40Þ

account for the REW-scale intra-subregion entropyexchange between thea-phase and the remaining phases.

APPENDIX B. CONSERVATION EQUATIONS FORTHE SATURATED ZONE (S-SUBREGION)

The saturated zone (s-subregion) comprises the subsurfaceregion, which is saturated with water, as depicted in Fig. 4.The s-subregion exchanges water with the overlying

subregions: the unsaturated zone, the saturated overlandflow and the main channel reach. At the bottom, the satu-rated zone can be confined either by impermeable strata orcan interact with the deep groundwater reservoir. The con-tact surface between the s-subregion and the channel isdefined by the channel bed, which is carved into the soil,while the contact surface with the o-subregion is coincidentwith the saturated land surface. Interaction with the neigh-bouring REWs or the external world across the mantle sur-face A is also possible, as evident from Figs 3 and 4.

The saturated zone constitutes a two-phase system due tothe coexistence of water and soil matrix. The general bal-ance law for a thermodynamic property for the s-subregiona-phase is written as follows:

ddt

∫Vs

rwgsadV þ

∫AsA

nsA·[r(v ¹ wsA)w ¹ i]gsadAdt

þ

∫Aus

nsu·[r(v ¹ wus)w ¹ i]gsadAdt

þ∑

j ¼ o, r

∫Asj

nsj ·[r(v ¹ wsj)w ¹ i]gsadAdt

þ

∫As

bot

ns·[r(v ¹ wsbot)w ¹ i]gs

adAdt

þ

∫Ss

ab

nab·[r(v ¹ wab)w ¹ i]gsadS¹

∫Vs

rfgsadV

¼

∫Vs

GgsadV; a ¼ m,w ðB1Þ

where the contact surfaceAso (seepage face) is formed bythe saturated land surface, in immediate contact with thesheet of water forming the overland flow, andAsr is definedby the channel bed. Following a time averaging proceduresimilar to that outlined for the u-subregion, the first termbecomes:

12Dt

ddt

∫t þ Dt

t ¹ Dt

∫Vs

rwgsadVdt ¼

ddt

(esaysqs〈r〉saws

aS) (B2)

Furthermore, the following REW-scale exchange terms canbe defined:

esAa ¼

12DtS

∫t þ Dt

t ¹ Dt

∫AsA

nsA·[r(wsA ¹ v)]gsadAdt; a ¼ m, w

(B3)

and

IsAa ¼

12DtS

∫t þ Dt

t ¹ Dt

∫AsA

nsA·[i ¹ r(v ¹ wsA)wsa]gs

adAdt

a ¼ m,w ðB4Þ

are the REW-scale mass exchange and the non-convectiveinteraction of thea-phase across the mantle surfaceA, wherethe deviation quantityws

a is defined in analogy to eqn (A6).Similar to what we observed for the u-subregion, the eqns(B3) and (B4) can be written as a sum of componentsrelative to each segment forming part of the mantle (see


eqn (2)):

euAa ¼

∑l

euAa, l þ euA

a,ext (B5)

IuAa ¼

∑l

IuAa, l þ IuA

a, ext (B6)

where the second term on the r.h.s. is non-zero only forREWs which have part of the mantle in common with theexternal watershed boundary. For all the following balanceequations the exchange across the mantle is implicitlyunderstood as a sum of more components. Next,

esja ¼

12DtS

∫t þ Dt

t ¹ Dt

∫Asj

nsj ·[r(wsj ¹ v)]guadAdt;

j ¼ u, o, r a ¼ m, w ðB7Þ

and

I sja ¼

12DtS

∫t þ Dt

t ¹ Dt

∫Asj

nsj ·[i ¹ r(v ¹ wsj)wsa]gs

adAdt

j ¼ u,o, r a ¼ m,w ðB8Þ

are the mass exchange and the non-convective interactionof the s-subregiona-phase with the unsaturated zone, withthe overland flow region and the channel, for solid andwater phases, respectively. In a similar fashion we define

es bota ¼

12DtS

∫t þ Dt

t ¹ Dt

∫As

bot

ns·[r(wsbot ¹ v)]gs

adAdt;

a ¼ m,w ðB9Þ

as the mass exchange term, and

I s bota ¼

12DtS

∫t þ Dt

t ¹ Dt

∫As

bot

ns·[i ¹ r(v ¹ wsbot)w

sa]gs

adAdt

a ¼ m,w ðB10Þ

as the non-convective exchange ofw between water andsolid phase of the saturated zone, the deep groundwater orthe underlying impermeable strata. Finally,

esab ¼

12DtS

∫t þ Dt

t ¹ Dt

∫Ss

ab

nab·[r(wab ¹ v)]gsadS; a,b¼ m,w

(B11)

is the mass exchange between thea-phase and theb-phase,while

I sab ¼

12DtS

∫t þ Dt

t ¹ Dt

∫Ss

ab

nab·[i ¹ r(v ¹ wab)wsa]gs

adSdt

a,b¼ m,w ðB12Þ

is the non-convective exchange ofw between the twophases. Substitution of the exchange terms into the eqn(B1), after it has been averaged in time, introduction ofaverage quantities given by eqns (5)–(8) and (15), and,finally, definition of REW-scale quantities based on the

averages, lead to the generica-phase balance equation:

ddt

(rsaws

aysesaqs) ¹ (esA

a þ IsAa ) ¹ rf s

aysesaqs

¼ Gsaqs þ (es bot

a wsa þ Is bot

a ) þ ðB13Þ∑j ¼ u, o, r

(esjaws

a þ I sja ) þ (es

abwsw þ I s

ab); a ¼ s,w

In the following paragraphs the conservation equations formass, momentum, energy and entropy will be stated. Theconcepts so far are straightforward extensions of what hasbeen shown for the u-subregion and, therefore, details willbe omitted.

Conservation of mass

The mass conservation for for the s-subregiona-phase isobtained from eqn (B13), analogous to what has been pur-sued for the u-subregion. The result is:

ddt

(rsayses

aqs) ¼ esAa þ es bot

a þ∑

j ¼ u,o, resja ; a,b ¼ m, w

(B14)

The terms on the r.h.s. represent the mass exchange of thea-phase with the neighbouring REWs and the externalworld acrossA, the exchanges with the deep groundwaterreservoir as well as the neighbouring u-, o- and r-subre-gions, respectively.

Conservation of momentum

The REW-scale equation for conservation of momentumis once again obtained from eqn (B13) by introducing theappropriate microscopic properties from Table 2 into eqn(B1):

ddt

(rsavs

aysesaqs) ¹ (esA

a vsa þ TsA

a ) ¹ rsags

aysesaqs

¼ (es bota vs

a þ Ts bota ) þ

∑j ¼ u,o, r

(esjavs

a þ Tsja)

þ (esabvs

a þ Tsab) ðB15Þ

where:

TsAa ¼

12DtS

∫t þ Dt

t ¹ Dt

∫AsA

nsA·[t ¹ r(v ¹ wsA)vsa]gs

adAdt

(B16)

is the momentum exchange between the s-subregiona-phase and neighbouring REWs as well as the externalworld across the mantle, and

Ts bota ¼

12DtS

∫t þ Dt

t ¹ Dt

∫As

bot

ns·[t ¹ r(v ¹ wsbot)vs

a]gsadAdt

(B17)

is the momentum exchange between the s-subregion andthe underlying impermeable strata or the deeper


groundwater. Furthermore,

Tsja ¼

12DtS

∫t þ Dt

t ¹ Dt

∫Auj

nsj ·[t ¹ r(v ¹ wsj)vsa]gs

adAdt;

j ¼ u,o, r ðB18Þ

are the momentum exchange terms between thes-subregion, the u-subregion, the overland flow region andthe channel reach, and

Tsab ¼

12DtS

∫t þ Dt

t ¹ Dt

∫Ss

ab

nab·[t ¹ r(v ¹ wab)vsa]gs

adSdt;

a,b¼ m,w ðB19Þ

accounts for the momentum exchange of the s-subregiona-phase with theb-phase across the intra-subregion phaseboundaries.

Conservation of energy

Thea-phase equation for conservation of energy is statedin analogy to what has been pursued for the u-subregion asfollows:

ddt

{rsa[Es

a þ (vsa)2=2]yses

aqs} ¹ { esAa [Es

a þ (vsa)2=2]

þ TsAa ·vs

a þ QsAa } ¹ ðB20Þ

rsa(gs

a·vsa þ hs

a)ysesaqs ¼ { es bot

a [Esa þ (vs

a)2=2]

þ Ts bota ·vs

a þ Qs bota } þ∑

j ¼ a,o, r{ esj

a [Esa þ (vs

a)2=2] þ Tsja·vs

a þ Qsja}

þ { esab[Es

a þ (vsa)2=2] þ Ts

ab·vsa þ Qs

ab}

where the REW-scale internal energy and external energysupply are given by

Esa ¼ Es

a þ (vsa)2s

=2 (B21)

hsa ¼ hs

a þ gsa·vs

as

(B22)

Next, the REW-scale energy exchange terms areintroduced:

QsAa ¼

12DtS

∫t þ Dt

t ¹ Dt

∫AsA

nsA·{ q þ t·vsa ¹ r(v ¹ wsA)

3 [Esa þ (vs

a)2=2]}gsadAdt ðB23Þ

is the energy transfer from the s-subregiona-phase towardsthe neighbouring REWs and the external world across themantle, while

Qs bota ¼

12DtS

∫t þ Dt

t ¹ Dt

∫As

bot

ns·{ q þ t·vsa ¹ r(v ¹ ws

bot)

3 [Esa þ (vs

a)2=2]}gsadAdt ðB24Þ

is the energy transfer from the s-subregion towards theunderlying deep groundwater or the impermeable strata.Furthermore, the term defined as

Qsja ¼

12DtS

∫t þ Dt

t ¹ Dt

∫Asj

nsj ·{ q þ t·vsa ¹ r(v ¹ wsj)

3 [Esa þ (vs

a)2=2]}gsadAdt; j ¼ u,o, r ðB25Þ

is the energy transfer from the s-subregiona-phase into theu-subregion, the overland flow region and the channelreach, and finally,

Qsab ¼

12DtS

∫t þ Dt

t ¹ Dt

∫Ss

ab

nab·{ q þ t·vsa ¹ r(v ¹ wab)

3 [Esa þ (vs

a)2=2]}gsadSdt; a,b¼ m,w ðB26Þ

accounts for the energy exchange of the s-subregiona-phase with theb-phase across the intra-subregion phaseboundary.

Balance of entropy

The balance equation for the s-subregiona-phase entropyis:

ddt

(rsayses

ahsaqs) ¹ (esA

a hsa þ FsA

a ) ¹ rsabs

aysesaqs ¼ Ls

aqs

þ (es bota hs

a þ Fs bota ) þ

∑j ¼ u,o, r

(esjahs

a þ Fsja )

þ (esabh

sa þ Fs

ab) ðB27Þ

where:

hsa ¼ hs

a (B28)

bsa ¼ bs

a (B29)

Lsa ¼ 〈L〉sa (B30)

are the REW-scale entropy, entropy supply and internalgeneration of entropy, respectively, while

FsAa ¼

12DtS

∫t þ Dt

t ¹ Dt

∫AsA

nsA·[j ¹ r(v ¹ wsA)hsa]gs

adAdt

(B31)

is the entropy exchange from the s-subregiona-phaseacross the mantle. Next,

Fs bota ¼

12DtS

∫t þ Dt

t ¹ Dt

∫As

bot

ns·[j ¹ r(v ¹ wsbot)hs

a]gsadAdt

(B32)

is the entropy exchange between the s-subregiona-phaseand the underlying deep groundwater or the impermeablelayers, while

Fsja ¼

12DtS

∫t þ Dt

t ¹ Dt

∫Asj

nsj ·[j ¹ r(v ¹ wsj)hsa]gs

adAdt;

j ¼ u,o, r ðB33Þ

is the entropy exchange between the s-subregiona-phase


and the u-subregion, the overland flow region and the chan-nel reach. Finally,

Fsab ¼

12DtS

∫t þ Dt

t ¹ Dt

∫Ss

ab

nab·[j ¹ r(v ¹ wab)hsa]gs

adSdt;

a,b¼ m,w ðB34Þ

accounts for the entropy exchange of the s-subregiona-phase with theb-phase across the intrasubregion phaseboundaries.

APPENDIX C. CONSERVATION EQUATIONS FORTHE REGION OF CONCENTRATED OVERLANDFLOW (C-SUBREGION)

The subregion here identified as concentrated overland flowzone, comprises all the flow occurring on the unsaturatedportion of the land surface. It includes Hortonian overlandflow and flow in rills and gullies. Furthermore, the c-sub-region comprises the sub-REW scale channel networkwithin the unsaturated land surface. This allows to accountfor the self-similar structure of the channel network, whichimplies that, whatever the scale of observation, there isalways a treelike branching network present at smallerscales. The flow within the c-subregion is supposed toinclude only the water phase. The presence of sedimenttransport phenomena isa priori excluded. The frameworkpresented here, however, does not necessarily require thisassumption, which is merely a simplifying expedient. Sedi-ment transport can be included by introducing a solid phasein the c-, o- and r-subregions in an appropriate fashion.

The c-subregion will be described as a sheet of waterwhich communicates with the atmosphere on top, the unsa-turated zone at the bottom, and with the saturated overlandflow sheet along the intersection line of the water table withthe land surface. The global balance law for the c-subregioncan be stated, in analogy to what has been pursuedpreviously:

ddt

∫Vc

rwdV þ

∫Acu

ncu·[r(v ¹ wcj)w

þ

∫Aco

nco·[r(v ¹ wco)w ¹ i]dAdt

þ

∫Ac

top

nc·[r(v ¹ wctop)w ¹ i]dAdt ¹

∫Vc

rf dV

¼

∫Vc

GdV ðC1Þ

where Aco is the sum of cross-sectional areas where theconcentrated overland flow merges with the saturated over-land flow sheet (see Fig. 4). After applying a similar time-averaging procedure to eqn (C1) as outlined in eqn (A2) forthe u-subregion, and employing the average quantities(eqns (5)–(7) and (15)), the first term becomes:

12Dt

ddt

∫t þ Dt

t ¹ Dt

∫Vc

rwdVdt ¼ddt

(ycqc〈a〉cwcS) (C2)

Next, appropriate REW-scale exchange terms for the prop-erty w have to be defined:

ecj ¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Acj

ncj ·[r(wcj ¹ v)]dAdt; j ¼ u,o

(C3)

are the mass exchange, and

I cj ¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Acj

ncj ·[i ¹ r(v ¹ wcj)wc]dAdt; j ¼ u,o

(C4)

are the non-convective fluxes between the c-subregion, theunsaturated zone across the areaAcu and the region ofsaturated overland flow. The deviationwc from the spaceand time average of the propertyw is defined as:

wc ¼ w ¹ wc (C5)

Subsequently,

ec top¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Ac

top

nc·[r(wctop ¹ v)]dAdt (C6)

is the mass exchange between the c-subregion and theatmosphere on top. The non-convective flux is consistentlydefined as:

Ic top¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Ac

top

nc·[i ¹ r(v ¹ wctop)wc]dAdt

(C7)

Substitution of the previously defined quantities into thetime-averaged general balance law (eqn (C1)), introductionof the averages defined through eqns (5)–(7) and (15) anddivision by the surface area projectiono yields:

ddt

(〈r〉cycwcqc) ¹ 〈r〉cycf cqc ¼ 〈G〉cqc þ (ec topwc þ Ic top)

þ (ecuwc þ I cu) þ (ecowo þ I co) ðC8Þ

The equation can, once again, be rewritten after definitionof REW-scale quantities in terms of averages:

ddt

(rcycwcqc) ¹ rcycf cqc ¼ Gcqc þ (ec topwc þ Ic top)

þ (ecuwc þ I cu) þ (ecowc þ Ico) ðC9Þ


The REW-scale mass balance for the c-subregion isobtained by introducing the appropriate microscopic


quantities from Table 2 into the balance eqn (C1). Theresulting mass balance assumes the following expression:

ddt

(rcycqc) ¼ ec topþ ecu þ eco (C10)

The terms on the r.h.s. account for the exchange of mass ofthe c-subregion with the atmosphere, with the underlying u-subregion and with the saturated overland flow region.


The REW-scale equation for conservation of momentumfor the c-subregion is derived b introducing the microscopicquantities according to Table 2 into eqn (C1):

ddt

(rcycvcqc) ¹ rcycgcqc ¼ (ec topvc þ Tc top)

þ (ecovc þ Tco) þ (ecuvc þ Tcu) ðC11Þ

where:

Tc top¼1

2DtS

∫t þDt

t ¹Dt

∫Ac

top

nc·[t ¹ r(v ¹ wctop)vc]dAdt

(C12)

is the REW-scale momentum exchange term between the c-subregion and the atmosphere, and

Tcj ¼1

2DtS

∫t þDt

t ¹Dt

∫Acj

ncj ·[t ¹ r(v ¹ wcj)vc]dAdt; j ¼ u, o

(C13)

is the momentum transfer to the u- and o-subregions.


The equation for conservation of energy is derived bydefining the microscopic quantities according to Table 2.The resulting equation is

ddt

{rcyc[Ec þ (vc)2=2]qc} ¹ rcyc(gc·vc þ hc)qc ¼

(C14)

{ ec top[Ec þ (vc)2=2] þ Tc top·vc þ Qc top}

þ { ecu[Ec þ (vc)2=2] þ Tcu·vc þ Qcu}

þ { eco[Ec þ (vc)2=2] þ Tco·vc þ Qco}

where:

Ec ¼ Ecþ (vc)2c

=2 (C15)

hc ¼ hcþ gc·vcc (C16)

are the REW-scale internal energy and the total heat supplyfrom the external world, respectively. Furthermore,

Qc top¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Ac

nc·{ q þ t·vc ¹ r(v ¹ wctop)

3 [Ecþ (vc)2=2]}dAdt ðC17Þ

is the exchange of energy with the atmosphere, and

Qcj ¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Acj

ncj ·{ q þ t·vc ¹ r(v ¹ wcj)

3 [Ecþ (vc)2=2]}dAdt; j ¼ u, o ðC18Þ

accounts for the energy transfer to the underlying unsatu-rated zone and the saturated overland flow region.

Balance of entropy

The REW-scale balance of entropy for the c-subregion isobtained by defining the microscopic quantities as given inTable 2. The equation resulting from eqn (C1) is:

ddt

(rcychcqc) ¹ rcycbcqc ¼ Lcqc þ (ec tophc þ Fc top)

þ (ecuhc þ Fcu) þ (ecohc þ Fco) ðC19Þ

where:

hc ¼ hc (C20)

bc ¼ bc (C21)

Lc ¼ 〈L〉c (C22)

are the entropy, and the REW-scale terms of entropy supplyand internal generation of entropy, respectively, while

Fc top¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Ac

nc top·[j ¹ r(v ¹ wctop)hc]dAdt

(C23)

is the term of entropy exchange between the c-subregionand the atmosphere, and

Fcj ¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Acj

ncj ·[j ¹ r(v ¹ wcj)hc]dAdt; j ¼ u, o

(C24)

account for the entropy exchange between the c-subregionand the underlying u-subregion and the overland flowregion, respectively.

APPENDIX D. CONSERVATION EQUATIONS FORTHE REGION OF SATURATED OVERLAND FLOW(O-SUBREGION)

The flow characterizing the the o-subregion is occurring onthe land surface, along the seepage faces of the saturatedzone (s-subregion). it also includes the sub-REW-scalechannel network flow within the saturated portion of theland surface. The saturated zone is composed of the waterand the solid phases. The exchange of the propertyw of theo-subregion with both phases of the underlying aquifer has,therefore, to be considered. The o-subregion can furtherexchangew with the main channel reach along the channeledge and with the the c-subregion from the uphill regions, asshown in Fig. 4. There is no interaction with the unsaturatedzone (u-subregion). Also here, presence of sediment transportis excluded. The generic balance law for the o-subregion canbe stated in analogy to what has been pursued previously:


ddt

∫Vo

rwdV þ

∫Ao

top

no·[r(v ¹ wotop)w ¹ j ]dAdt

þ

∫Aso

nos·[r(v ¹ wos)w ¹ j ]dAdt

þ

∫Aco

noc·[r(v ¹ wco)w ¹ j ]dAdt

þ

∫Aor

nor·[r(v ¹ wor)w ¹ j ]dAdt ¹

∫Vo

rf dV

¼

∫Vo

GdV ðD1Þ

whereAor is the total cross-sectional area of the overlandflow water sheet at the inflows along the channel edges.After application of an analogous procedure as pursued forthe unsaturated zone (see eqn (A2)), and by employing theaverage quantities defined in Appendix E, the first term ofeqn (D1) becomes:

12Dt

ddt

∫t þ Dt

t ¹ Dt

∫Vo

rwdVdt ¼ddt

(yoqo〈r〉owoS) (D2)

Next, we introduce appropriate REW-scale exchange termsfor the propertyw:

eo top¼1

2DtS

∫t þDt

t ¹Dt

∫Ao

top

no·[r(wotop ¹ v)]dAdt (D3)

is the exchange between the o-subregion and the atmos-phere on top, and

Io top¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Ao

top

no·[j ¹ r(v ¹ wotop)wo]dAdt

(D4)

is the respective non-convective flux term, where thedeviation quantitywo is defined in analogy to eqn (C5).Furthermore,

eoj ¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Aoj

noj ·[r(woj ¹ v)]dAdt; j ¼ s, c, r

(D5)

are the mass exchange terms between the o-subregion andthe underlying s-subregion across the seepage faceAos,with the c-subregion from uphill, and the channel reachthrough lateral inflow. The non-convective fluxes aredefined as:

Ioj ¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Aoj

noj ·[j ¹ r(v ¹ woj)wo]dAdt; j ¼ s, c, r

(D6)

Substitution of the previously defined quantities into eqn(D1) after it has been averaged in time, introduction of theaverage quantities eqns (5)–(7) and (15), and division bythe surface area projectiono yields:

ddt

(〈r〉oyowrqo) ¹ 〈r〉oyof oqo ¼ 〈G〉oqo þ (eo topwoIo top)

(D7)þ (eoswo þ Ios) þ (eocwo þ Ioc) þ (eorwo þ Ior)

The equation can be restated in terms of appropriate REW-scale quantities:

ddt

(royowoqo) ¹ royof oqo ¼ Goqo þ (eo topwo þ Io top)

þ (eoswo þ Ios) þ (eocwo þ Ioc) þ (eorwo þ Ior) ðD8Þ


The REW-scale mass balance for the o-subregion isobtained by introducing the corresponding microscopicquantities from Table 2 into the balance eqn (D1). Theresulting mass balance eqn (D8) assumes the followingexpression:

ddt

(royoqo) ¼ eo topþ eosþ eoc þ eor (D9)

The terms on the r.h.s. account for the exchange of mass of theo-subregion with the atmosphere, with the s- and c-sub-regions, and with the channel reach through lateral inflow.


The REW-scale equation for conservation of momentumfor the o-subregion is derived by introducing the micro-scopic quantities according to Table 2 into eqn (D1). Theequation resulting from eqn (D2) is:

ddt

(royovoqo) ¹ royogoqo ¼ (eo topvo þ To top)

þ (eosvo þ Tos) þ (eocvo þ Toc) þ (eorvo þ Tor) ðD10Þ

where:

To top¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Ao

top

no·[t ¹ r(v ¹ wotop)vo]dAdt

(D11)

is the momentum exchange term between the o-subregionand the atmosphere, while

Toj ¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Aoj

noj ·[t ¹ r(v ¹ woj)vo]dAdt;

j ¼ s,c, r ðD12Þ

are the momentum transfer terms to the s-subregion, thec-subregion and the channel reach, respectively.


The equation for conservation of energy is derived bydefining the microscopic quantities according to Table 2.


The resulting equation isddt

{royo[Eo þ (vo)2=2]qo} ¹ royo(go·vo þ ho)qo

¼ { eo top[Eo þ (vo)2=2] þ To top·vo þ Qo top}

þ { eos[Eo þ (vo)2=2] þ Tos·vo þ Qos}

þ { eoc[Eo þ (vo)2=2] þ Toc·vo þ Qoc}

þ { eor[Eo þ (vo)2=2] þ Tor·vo þ Qor} ðD13Þ

where:

Eo ¼ Eoþ (vo)2o

=2 (D14)

ho ¼ hoþ go·voo

(D15)

are the average internal energy and the total heat supplyfrom the external world, respectively,

Qo top¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Ao

no·{ q þ t·vo ¹ r(v ¹ wotop)

3 [Eoþ (vo)2=2]}dAdt ðD16Þ

is the energy exchange with the atmosphere, and

Qoj ¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Aoj

noj ·{ q þ t·vo ¹ r(v ¹ woj)

3 [Eoþ (vo)2=2]}dAdt; j ¼ s, c, r ðD17Þ

are the energy transfer terms to the underlying saturatedzone, the concentrated overland flow and the channel reach.

Balance of entropy

The REW-scale balance of entropy for the o-subregion isobtained by defining the microscopic quantities as given inTable 2. The equation resulting from eqn (D1) is:

ddt

(royohoqo) ¹ royoboqo ¼ Loqo þ (eo topho þ Fo top)

þ (eosho þ Fos) þ (eocho þ Foc) þ (eorho þ For) ðD18Þ

where:

ho ¼ ho (D19)

bo ¼ bo (D20)

Lo ¼ 〈L〉o (D21)

are the REW-scale entropy and the terms of entropy supplyand internal generation of entropy, respectively, while

Fo top¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Ao

no top·[j ¹ r(v ¹ wotop)ho]dAdt

(D22)

is the term of entropy exchange between the o-subregion

and the atmosphere, and

Foj ¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Aoj

noj ·[j ¹ r(v ¹ woj)ho]dAdt

j ¼ s,c, r ðD23Þ

are the entropy exchange terms between the o-subregion,the underlying s-subregion, the c-subregion from the uphillregions, and the channel reach.

APPENDIX E. CONSERVATION EQUATIONS FORTHE CHANNEL REACH (R-SUBREGION)

The main channel reach of the REW exchangesw with theatmosphere on the channel free surface, with the underlyingsaturated zone (s-subregion) across the channel bed, withthe overland run-off areas (o-subregion) along the edges ofthe channel, with the neighbouring REWs and the externalworld across the mantleA at the REW outlet and inlet. Werecall that, in the case of REWs associated withfirst orderstreams, there is only an outlet and no inlet. The geometricproperties inherent to the channel at a cross-section are thewidth of the free surface, the wetted perimeter, and thecross-sectional aream (equivalent to a volume per unitlength [L3/L]) normal to the spatial curveCr forming theaxis of the channel. The volumeVr associated with thechannel reach is slender and can be approximated throughthe integration

Vr ¼

∫C r

mdC (E1)

where dC is an infinitesimal segment of the curveCr. Bymaking this approximation, the effects of volume distortiondue to curvature of the channel have been neglected. Therespective terms have been derived in a rigorous manner byGray et al.,15 to which the reader is referred for moredetailed explanation. The general conservation equationfor a generic propertyw within the volumeVr is stated as:

ddt

∫Vr

rwdV þ

∫ArA

nrA·[r(v ¹ wrA)w ¹ i]dAdt þ (E2)

∫Ar

top

nr·[r(v ¹ wrtop)w ¹ i]dAdt

þ

∫Asr

nrs·[r(v ¹ wsr)w ¹ i]dAdt þ

∫Aor

nro·[r(v ¹ wor)w ¹ i]dAdt ¹

∫Vr

rf dV ¼

∫Vr

GdV

whereArA is the total cross-sectional area defined by theintersection of the channel with the mantleA at the outletand inlet, andAr

top is the channel free surface. After appli-cation of the top time-averaging theorem (eqn (A2)), asshown for the case of the unsaturated zone, and use ofthe average quantities eqns (12)–(14) and (16), defined in


Appendix 5, the first term of eqn (E2) becomes:

12Dt

ddt

∫t þ Dt

t ¹ Dt

∫Vr

rwdVdt ¼ddt

(mryr〈r〉rwrS) (E3)

The following exchange terms between the channel reachand its surroundings are subsequently defined:

erA ¼1

2DtS

∫t þ Dt

t ¹ Dt

∫ArA

nrA·[r(wrA ¹ v)]dAdt (E4)

is the mass exchange of the channel reach across the mantleat the outlet and inlet, and

I rA ¼1

2DtS

∫t þ Dt

t ¹ Dt

∫ArA

nrA·[i ¹ r(v ¹ wrA)wr]dAdt (E5)

is the non-convective interaction of the channel acrossA,where, once again, the deviationwr from the time and spaceaverage is defined as:

wr ¼ w ¹ wr (E6)

We observe that the exchanges across the inlet and outletsections can be separated into a number of components. Ifthe REW is relative to afirst order stream, the exchangeacross the REW mantle occurs only at the outlet. If theREW is relative to ahigher order stream, there are tworeaches converging at the inlet and there is a reach follow-ing further downstream at the REW outlet, i.e. the channelreach is communicating with the reaches of three neigh-bouring REWs. For example, with reference to Fig. 1(c),the channel reach of REW 5 is communicating with thereaches of REWs 3 and 4 at the inlet and with the reachof REW 7 at the outlet. In addition, the REW, which is theclosest to the outlet, can interact across the externalwatershed boundary. With these considerations in mind,we rewrite eqns (E4) and (E5) in a general form:

erA ¼∑

l

erAl þ erA

ext (E7)

I rA ¼∑

l

I rAl þ I rA

ext (E8)

where the summation extends over the neighbouring REWs(three in the case of ahigher order streamand one in thecase of afirst order stream) and the second term on ther.h.s. is non-zero only for the REW next to the watershedoutlet. In all the following balance equations the exchangeterm across the mantle is implicitly understood as a sum ofthese components. Next,

er top ¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Ar

top

nr·[r(wrtop ¹ v)]dAdt (E9)

is the REW-scale mass exchange between the channel freesurface and the atmosphere, and

I r top ¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Ar

top

nr·[i ¹ r(v ¹ wrtop)w

r]dAdt

(E10)

is the-respective non-convective flux term. Finally,

erj ¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Aor

nrj ·[r(wrj ¹ v)]dAdt; j ¼ s,o

(E11)

is the mass exchange term of the channel reach with thesaturated zone across the channel bed and with the o-sub-region through lateral inflow, while

I rj ¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Arj

nrj ·[i ¹ r(v ¹ wrj)wr]dAdt; j ¼ s,o

(E12)

is the non-convective interaction between the channelreach, the saturated zone and the o-subregion. Introductionof the previous defined quantities into the time-averagedbalance law (eqn (E2)) and use of the definitions givenby eqns (12)–(14) and (16), yields:

ddt

(〈r〉rmrwryrS) ¹ (erAwr þ I rA)S ¹ 〈r〉rmr f ryrS

¼ 〈G〉ryrS þ (er topwr þ I r top)S þ (erswr þ I rs)S

þ (erowr þ I ro)S ðE13Þ

The use of REW-scale quantities, defined on the basis ofaverages, allows, after division byo, to recast the generalbalance equation for the channel reach:

ddt

(rrmrwryr) ¹ (erAwr þ I rA) ¹ rrmrf ryr

¼ Gryr þ (er topwr þ I r top) þ (erswr þ I rs)

þ (erowr þ I ro) ðE14Þ

In the following sections appropriate quantities will beintroduced into eqn (E14), in order to state the balancelaws for the four fundamental properties of mass, momen-tum, energy and entropy.


For the mass conservation along the spatial curveCr, themicroscale properties in eqn (E2) have to be chosenamongst the appropriate values for the mass balance fromTable 2. The REW-scale r-subregion mass balance assumesaccording to eqn (E14) the following expression:

ddt

(rrmryr) ¹ erA ¼ er top þ ers þ ero (E15)

The second term on the l.h.s. accounts for the exchange ofwater between the channel reach and the neighbouringREWs as well as the external world across A (i.e. inflowand outflow discharge), whereas the terms on the r.h.s.represent the mass exchange with the atmosphere at thefree surface (i.e. rainfall and open water evaporation),with the adjacent aquifer across the channel bed (i.e.recharge from groundwater) and with the overland flowregion along the channel edge (i.e. lateral inflow).



The REW-scale equation for conservation of momentumfor the r-subregion is obtained after defining the-microscaleproperties according to Table 2:

ddt

(rrmrvryr) ¹ (erAvr þ TrA) ¹ rrmrgryr

¼ (er topvr þ Tr top) þ (ersvr þ Trs) þ (erovr þ Tro)

ðE16Þ

where:

TrA ¼1

2DtS

∫t þ Dt

t ¹ Dt

∫ArA

nrA·[t ¹ r(v ¹ wrA)vr]dAdt

(E17)

is the REW-scale momentum exchange term withthe neighbouring REWs as well as the external worldacross the outlet and inlet sections on the mantleA.Furthermore,

Tr top ¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Ar

top

nr·[t ¹ r(v ¹ wrtop)vr]dAdt

(E18)

is the momentum transfer into the atmosphere on the chan-nel free surface, and

Trj ¼1

2DtS

∫t þDt

t ¹Dt

∫Arj

nrj ·[t ¹ r(v ¹ wrj)vr]dAdt; j ¼ s,o

(E19)

are the REW-scale terms of momentum exchangewith the saturated zone across the channel bed and withthe regions of saturated overland flow along the channeledges.


The equation for conservation of energy for the channelreach is given, after the appropriate substitutions for themicroscopic properties, by the expression:

ddt

{rrmr[Er þ (vr)2=2]yr} ¹ { erA[Er þ (vr)2=2]

þ TrA·vr þ QrA} ¹ rrmr(gr·vr þ hr)yr

¼ { er top[Er þ (vr)2=2] þ Tr top·vr þ Qr top}

þ { ers[Er þ (vr)2=2] þ Trs·vr þ Qrs} þ { ero[Er þ (vr)2=2]

þ Tro·vr þ Qro} ðE20)

where:

Er ¼ Erþ (vr)2r

=2 (E21)

is the REW-scale internal energy of the channel reach and

hr ¼ hrþ gr·vrr (E22)

is the REW-scale energy supply, consisting of the externalsupply in addition to the energy supply due to fluctuationsof velocity and gravity at the sub-REW-scale. Next, theREW-scale interaction terms are defined:

QrA ¼1

2DtS

∫t þ Dt

t ¹ Dt

∫ArA

nrA·{ q þ t·vr ¹ r(v ¹ wrA)

3 [Erþ (vr)2=2]}dAdt ðE23Þ

is the energy transfer from the channel reach across themantle at the outlet and inlet cross-sections,

Qr top ¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Ar

nr·{ q þ t·vr ¹ r(v ¹ wrtop)

3 [Erþ (vr)2=2]}dAdt ðE24Þ

is the energy exchange with the atmosphere at the channelfree surface, and

Qrj ¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Arj

nrj ·{ q þ t·vr ¹ r(v ¹ wrj)

3 [Erþ (vr)2=2]}dAdt; j ¼ s,o ðE25Þ

are the energy transfer terms into the s-subregion across thechannel bed, and the saturated overland flow region alongthe channel edges.

Balance of entropy

The balance of entropy for the channel reach is given,after appropriate substitution for respective microscopicquantities into eqn (E2), by the expression:

ddt

(rrmrhryr) ¹ (erAhr þ FrA) ¹ rrmrbryr

¼ Lryr þ (er tophr þ Fr top) þ (ershr þ Frs) þ (erohr þ Fro)

ðE26Þ

where:

hr ¼ hr (E27)

br ¼ br (E28)

Lr ¼ 〈L〉r (E29)

are the REW-scale entropy, and the REW-scale terms forentropy supply and internal generation of entropy, respec-tively, while the entropy exchange terms,are defined asfollows:

FrA ¼1

2DtS

∫t þDt

t ¹Dt

∫ArA

nrA·[j ¹ r(v ¹ wrA)hr]dAdt

(E30)is the entropy exchange with the neighbouring REWs andthe external world across the mantle A at the outlet andinlet, while

Fr top ¼1

2DtS

∫t þDt

t ¹Dt

∫Ar

nr·[j ¹ r(v ¹ wrtop)h

r]dAdt

(E31)


is the entropy exchange with the atmosphere across thechannel free surface. Finally, the term defined as

Frj ¼1

2DtS

∫t þ Dt

t ¹ Dt

∫Arj

nrj ·[j ¹ r(v ¹ wrj)hr]dAdt; j ¼ s,o

(E32)

expresses the entropy exchange of the channel reach withthe s- and the o-subregions.


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