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A unifying framework for watershed thermodynamics: constitutiverelationships

Paolo Reggiani a, S. Majid Hassanizadeh b, Murugesu Sivapalan a, William G. Gray a,*,1

a Centre for Water Research, Department of Environmental Engineering, The University of Western Australia, 6907 Nedlands, Australiab Section of Hydrology and Ecology, Faculty of Civil Engineering and Geosciences, Delft University of Technology, P.O. Box 5048, 2600GA Delft,

The Netherlands

Accepted 5 February 1999

Abstract

The balance equations for mass and momentum, averaged over the scale of a watershed entity, need to be supplemented with

constitutive equations relating ¯ow velocities, pressure potential di�erences, as well as mass and force exchanges within and across

the boundaries of a watershed. In this paper, the procedure for the derivation of such constitutive relationships is described in detail.

This procedure is based on the method pioneered by Coleman and Noll through exploitation of the second law of thermodynamics

acting as a constraint-type relationship. The method is illustrated by its application to some common situations occurring in real

world watersheds. Thermodynamically admissible and physically consistent constitutive relationships for mass exchange terms

among the subregions constituting the watershed (subsurface zones, overland ¯ow regions, channel) are proposed. These consti-

tutive equations are subsequently combined with equations of mass balance for the subregions. In addition, constitutive relation-

ships for forces exchanged amongst the subregions are also derived within the same thermodynamic framework. It is shown that,

after linearisation of the latter constitutive relations in terms of the velocity, a watershed-scale Darcy's law governing ¯ow in the

unsaturated and saturated zones can be obtained. For the overland ¯ow, a second order constitutive relationship with respect to

velocity is proposed for the momentum exchange terms, leading to a watershed-scale Chezy formula. For the channel network

REW-scale Saint±Venant equations are derived. Thus, within the framework of this approach new relationships governing exchange

terms for mass and momentum are obtained and, moreover, some well-known experimental results are derived in a rigorous

manner. Ó 1999 Elsevier Science Ltd. All rights reserved.

Keywords: Watersheds; Hydrologic theory; Second law of thermodynamics; Mass and force balances; Constitutive relationships; Coleman and Noll

procedure

1. Introduction

This work represents the sequel to a previous paper(Reggiani et al. [33]) concerned with the derivation ofwatershed-scale conservation equations for mass, mo-mentum, energy and entropy. These equations havebeen derived by averaging the corresponding point scalebalance equations over a well de®ned averaging regioncalled the Representative Elementary Watershed(REW). The REW is a fundamental building block forhydrological analysis, with the watershed being disc-retised into an interconnected set of REWs, where thestream channel network acts as a skeleton or organising

structure. The stream network associated with a water-shed is a bifurcating, tree-like structure consisting ofnodes inter-connected by channel reaches or links. As-sociated with each reach or link, there is a well-de®nedarea of the land surface capturing the atmospheric pre-cipitation and delivering it towards the channel reach.These areas uniquely identify the sub-watersheds whichwe de®ne as REWs. As a result, the agglomeration ofthe REWs forming the entire watershed resembles thetree-like structure of the channel network on which thediscretisation is based, as shown schematically in Fig. 1.

The volume making up a REW is delimited externallyby a prismatic mantle, de®ned by the shape of the ridgescircumscribing the sub-watershed. On top, the REW isdelimited by the atmosphere, and at the bottom by ei-ther an impermeable substratum or an assumed limitdepth. The stream reach associated with a given REWcan be either a source stream, classi®ed as a ®rst order

Advances in Water Resources 23 (1999) 15±39

* Corresponding author. Fax: +61 8 9387 8211; e-mail: paolo.regg-

[email protected] On leave from the Department of Civil Engineering and Geological

Sciences, University of Notre Dame, Notre Dame, IN 46556, USA.

0309-1708/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved.

PII: S 0 3 0 9 - 1 7 0 8 ( 9 9 ) 0 0 0 0 5 - 6

stream by Horton and Strahler [26,38], or can inter-connect two internal nodes of the network, in which caseit is classi®ed as a higher order stream.

The size of the REWs used for the discretisation ofthe watershed is determined by the spatial and temporal

resolutions, which are sought for the representation ofthe watershed and its response, as well as by the reso-lution of available data sets. Change of resolution isequivalent to a change of stream network density, whilestill maintaining the bifurcating tree-like structure.

Nomenclature

Latin symbolsA mantle surface with horizontal normal

delimiting the REW externallyA linearisation coe�cent for the mass

exchange terms, �T=L�A areal vector de®ned through Eq. (2.4)b external supply of entropy, �L2=T 3 o�B linearisation coe�cent for the mass

exchange terms, �M=L3�e mass exchange per unit surface area,

�M=TL2�ex; ey ; ez: unit vectors pointing along the x, y and z

axes, respectivelyE internal energy per unit mass, �L2=T 2�E extensive internal energy, �ML2=T 2�f external supply term for wF entropy exchange per unit surface area

projection, �M=T 3 o�g the gravity vector, �L=T 2�G production term in the generic balance

equationh external energy supply, �L2=T 3�i general ¯ux vector of wj microscopic non-convective entropy ¯ux,

�M=T 3 o�J rate of rainfall input or evaporation,

�M=L2T �L rate of net production of entropy per unit

area, �M=T 3 oL2�mr volume per unit channel length, equivalent

to the average cross sectional area, �L3=L�M number of REWs making up the entire

watershednij unit normal vector to the boundary

between subregion i and subregion jni

n unit normal vector to i-subregion average¯ow plane

nit unit tangent vector to i-subregion average

¯ow planeO the global reference systemp pressure, �F =L2�q heat vector, �M=T 3�Q energy exchange per unit surface area

projection, �M=T 3�R ®rst order friction term, �FT=L3�

U second order friction term, �FT 2=L4�s the saturation function, �ÿ�t microscopic stress tensor, �M=T 2L�T momentum exchange per unit surface area

projection, �M=T 2L�v velocity vector of the bulk phases, �L=T �V volume, �L3�w velocity vector for phase and subregion

boundaries, �L=T �yi average vertical thickness of the i-subre-

gion along the vertical, �L�wr average channel top width, �L�Greek symbolsci slope angle of the i-subregion ¯ow plane

with respect to the horizontal planeD indicates a time increment� porosity, �ÿ��ia i-subregion a-phase volume fraction, �ÿ�

g the entropy per unit mass, �L2=T 2 o�h the temperatureKco contour curve separating the two overland

¯ow regions from each otherKor contour curve forming the edge of the

channelnr the length of the main channel reach Cr per

unit surface area projection R, �1=L�q mass density, �M=L3�R projection of the total REW surface area S

onto the horizontal plane, �L2��s the non-equilibrium part of the momentum

exchange terms/ the gravitational potentialxi The time-averaged surface area fraction

occupied by the j-subregion, �ÿ�Subscripts and superscriptsi; j superscripts which indicate the various

phases or subregions within a REWk subscript which indicates the various

REWs within the watershedtop superscript for the atmosphere, delimiting

the domain of interest at the topbot superscript for the the region delimiting the

domain of interest at the bottoma;b indices which designate di�erent phasesm, w, g designate the solid matrix, the water and

the gaseous phase, respectively

16 P. Reggiani et al. / Advances in Water Resources 23 (1999) 15±39

Consequently, the way we have de®ned the REWs withrespect to the stream network assures the invariance ofthe concept of REW with change of spatial scale.

The ensemble of REWs constituting the watershedcommunicate with each other by way of exchanges of

mass, momentum and energy through the inlet andoutlet sections of the associated channel reaches. Inaddition, they can also communicate laterally throughexchanges of these thermodynamic properties across themantle separating them (through the soils). The REW-scale conservation equations are formulated by averag-ing the balance laws over ®ve subregions forming theREW, as depicted in Fig. 2. These subregions have beenchosen on the strength of previous ®eld evidence aboutdi�erent processes which operate within catchments,their ¯ow geometries and time scales. The ®ve subre-gions chosen in this work are denoted as follows: Un-saturated zone, Saturated zone, Concentrated overland¯ow, saturated Overland ¯ow and main channel Reach.The unsaturated and saturated zones form the subsur-face regions of the REW where the soil matrix coexistswith water (and the gas phase in the case of the unsat-urated zone). The concentrated overland ¯ow subregionincludes surface ¯ow within rills, gullies and smallchannels, and the regions a�ected by Hortonian over-land ¯ow. It covers the unsaturated portion of the landsurface within the REW. The saturated overland ¯owsubregion comprises the seepage faces, where the watertable intersects the land surface and make up the satu-rated portion of the REW land surface. For more in-depth explanations about the concept of REW thereader is referred to Reggiani et al. [33].

The REW-scale balance equations obtained by theaveraging procedure represent the various REWs asspatially lumped units. Hence these equations form a setof coupled non-linear ordinary di�erential equations(ODE), in time only; the only spatial variability allowedis between REWs. Any spatial variability at the

Fig. 1. Hierarchical arrangement of 13 REWs forming a watershed.

Fig. 2. Detailed view of the ®ve subregions forming a REW.

P. Reggiani et al. / Advances in Water Resources 23 (1999) 15±39 17

sub-REW-scale is averaged over the REW and can berepresented in terms of e�ective parameterisations in theconstitutive equations to be derived in this paper.

In the course of the averaging procedure a series ofexchange terms for mass, momentum, energy and en-tropy among phases, subregions and REWs have beende®ned. These terms are the unknowns of the problem.A major di�culty is that the total number of unknownsexceeds the number of available equations. The de®cit ofequations with respect to unknowns requires that anappropriate closure scheme has to be proposed, whichwill lead to the derivation of constitutive relationshipsfor the unknown exchange terms. In this paper we re-solve the closure problem by exploiting the second lawof thermodynamics (i.e., entropy inequality) as a con-straint-type relationship. This allows us to obtain ther-modynamically admissible and physically consistentconstitutive equations for the exchange terms within theframework of a single procedure, applied uniformly andconsistently across the REWs. This approach for thederivation of constitutive relationships is known in theliterature as Coleman and Noll [5] method.

The second law of thermodynamics constitutes aninequality representing the total entropy production of asystem. The inequality can be expressed in terms of thevariables and exchange terms for mass, momentum andenergy of the system and is subject to the condition ofnon-negativity. Furthermore, the entropy inequality issubject to a minimum principle, as explained, for ex-ample, by Prigogine [32]. An absolute minimum of en-tropy production is always seen to hold underthermodynamic equilibrium conditions. Consequently,under these circumstances, the entropy production iszero. In non-equilibrium situations the entropy in-equality has to assume always positive values, becausethe second law of thermodynamics dictates that theentropy production of the system is never negative. Thisimposes precise constraints on the functional form of theconstitutive parameterisations and reduces the degree ofarbitrariness in their choice.

The Coleman and Noll method has been successfullyapplied by Hassanizadeh and Gray for deriving consti-tutive relationships in the area of multiphase ¯ow[22,18], for ¯ow in geothermal reservoirs [17], for thetheoretical derivation of the Fickian dispersion equationfor multi-component saturated ¯ow [21] and for ¯ows inunsaturated porous media [23,19].

Next to the thermodynamic admissibility, a furtherguideline for the constitutive parameterisations is thenecessity of capturing the observed physical behaviourof the system, including ®eld evidence. For example, ithas been shown that overland and channel ¯ows obeyChezy-type relationships, where the momentum ex-change between water and soil is given by a second orderfunction of the ¯ow velocity. Similarly, according toDarcy's law, the ¯ow resistivities for slow subsurface

¯ow can be expressed as linear functions of the velocity.We will show that the proposed constitutive parame-terisations will lead, under steady state conditions, to aREW-scale Darcy's law for the unsaturated and thesaturated zones, and to a REW-scale Chezy formula forthe overland ¯ow. In the case of ¯ow in the channelnetwork an equivalent of the Saint±Venant equations fora bifurcating structure of reaches will be obtained.

The ®nal outcome of this paper is a system of 19 non-linear coupled ordinary di�erential equations in as manyunknowns for every REW. This set of equations needsto be solved simultaneously with the equation systemsgoverning the ¯ow in all the remaining REWs formingthe watershed. A coupled (simultaneous) solution of theequation systems is necessary, since the ¯ow ®eld in oneREW can in¯uence the ¯ow ®eld in neighbouring REWsthrough up- and downstream backwater e�ects alongthe channel network, and through the regionalgroundwater ¯ow crossing the REW boundaries. Indeveloping the constitutive theory presented in this pa-per, we will make a number of simplifying assumptionsto keep the problem manageable. Especially, the theoryfocusses on runo� processes at the expense of evapo-transpiration, and hence the treatment of the latter isless than complete. Thermal e�ects, e�ects of vapourdi�usion, vegetation e�ects and interactions with theatmospheric boundary layer will be neglected. These willbe left for further research.

2. Balance laws, second law of thermodynamics and

conditions of continuity

2.1. REW-scale balance laws

REW-scale conservation laws for mass, momentum,energy and entropy for the ®ve subregions occupying theREW have been derived rigorously by Reggiani et al.[33] for a generic thermodynamic property w. In addi-tion, the balance laws have been averaged in time, toaccommodate di�erent time scales associated with thevarious ¯ow processes occurring within the watershed.The generic conservation law for the a-phase (water,soil, gas) within the i-subregion of an REW can beformulated as follows:

1

2Dtd

dt

Zt�Dt

tÿDt

ZV ia

qwdV

�Xj 6�i

1

2Dt

Zt�Dt

tÿDt

ZAij

a

nij � �qw�vÿ wij� ÿ i�dA

� 1

2Dt

Zt�Dt

tÿDt

ZV ia

qf dV � 1

2Dt

Zt�Dt

tÿDt

ZV i

a

qGdV �2:1�


where f is the external supply of w, G is its rate ofproduction per unit mass, q is the mass density, v is thevelocity of the phase, wij is the velocity of the boundary,and Dt is an appropriate time averaging interval. Thevolume V i

a ; �L� indicates the volume ®lled by the i-sub-region a-phase, normalised with respect to the hori-zontal surface area projection of the REW, R. Thesurface Aij

a is the portion of boundary, delimiting the i-subregion a-phase towards the phase or subregion j. Aij

a

can assume the symbol Aus for the water table, Auc forthe unsaturated land surface, Aos for the seepage face, Asr

for the surface formed by the channel bed, Aoc or Aor toindicate the ¯ow cross sections forming the boundariesbetween the saturated and the concentrated overland¯ow or the saturated overland ¯ow and the channel,respectively. We note that the order of the superscriptscan be interchanged arbitrarily. The surfaces delimitingthe concentrated overland ¯ow, saturated overland ¯owor the channel towards the atmosphere, are indicatedwith Ao top, Ac top and Ar top, respectively. The portion ofmantle surface delimiting the unsaturated and the sat-urated zones laterally, are indicated with AuA and AsA.Finally, phase interfaces, such as the water±gas, water±soil and soil±gas interfaces are indicated with the sym-bols Swg, Sws and Ssg. All these surfaces are summarisedin Table 1.

We further observe that, in the case of the unsatu-rated zone, there are three phases present: the soil matrixindicated with m, the water phase w and the air±vapourmixture g. The saturated zone contains two phases, thewater and the soil matrix. The overland ¯ow zones andthe channel reach comprise only the water phase. Thevarious volumina V i

a , expressed on a per-unit-area basisR, can be expressed as products of respective geometricvariables. In the case of the unsaturated and saturatedzones, V i

a is the product of the area fractions xi; �ÿ�,occupied by the i-subregion, the average vertical thick-ness of the respective subregion, yi; �L�, and the a-phase

volume fraction, �ia. For the saturated and the concen-

trated overland ¯ow regions, the volume V i of the onlyphase present (water), is the product of the horizontalarea fraction xi and the average vertical thickness of the¯ow sheet, yi. Finally, in the case of the channel, V r isthe product of the length of the channel reach per unitarea, nr; �Lÿ1�, and the average channel cross sectionalarea, mr; �L2�. The notations for these quantities aresummarised in Table 1.

To obtain speci®c balance equations for mass, mo-mentum, energy and entropy, the corresponding mi-croscopic quantities indicated in Table 2 need to besubstituted into Eq. (2.1). We note that, according towhich type of balance equation we wish to obtain, thethermodynamic property, w, can be either equal to 1, orcan assume the symbol of the velocity, v, the sum of theinternal and kinetic energy, E � v2=2, or the entropy, g.The non-convective ¯ux, i, can be set equal to the zerovector, the stress tensor, t, the sum of the product t � vplus the heat ¯ux, q, or the entropy ¯ux, j. The externalsupply term of w, f , can be set equal to zero, to thegravity vector, g, the product g � v plus the energy sup-ply, h, or the external entropy supply, b. We observe thatthe gravity can be alternatively expressed as the gradientof the gravitational potential, /ÿ /0 � g�zÿ z0�, de-®ned with respect to a datum. This form of f will beneeded for the derivation of constitutive relationships.Finally, the internal production of w, G, is non-zero onlyfor the balance of entropy and is set equal to L.

In addition, watershed-scale exchange for mass,forces, heat and entropy are introduced, which accountfor the transfer of these quantities among phases, sub-regions and REWs. These have been accurately de®nedon a case-by-case basis by Reggiani et al. [33]. Fur-thermore, these exchange terms constitute unknownquantities of the problem, for which constitutive rela-tionships need to be sought. The following paragraphspresent the resulting expressions for the balance

Table 2

Summary of the properties in the conservation equations

Balance equation w i f G

Mass 1 0 0 0

Linear Momentum v t g or ÿr�/ÿ /0� 0

Energy E � 12v2 t � v� q g � v� h or ÿr�/ÿ /0� � v� h 0

Entropy g j b L

Table 1

Summary of the properties in the conservation equations

Subregion Index i Index a Boundaries Aij Volume V ia

Unsaturated zone u w,m,g Ausa ;A

uca ;A

uAa ; S

wm; Swg; Sgm �uayuxu

Saturated zone s w,m Ausa ;A

soa ;A

sAa ; S

wm �saysxs

Saturated overland ¯ow o None Aoc;Aso;Aor;Ao top yoxo

Concentrated overland ¯ow c None Aoc;Auc;Ac top ycxc

Channel reach r None ArA;Ars;Aro;Ar top mrnr


equations for mass, momentum, energy and entropy forthe a-phase contained within the i-subregion. The bal-ance equations are all expressed on a per-unit-area basisthrough division by the surface area projection R of theREW.

Conservation of mass. The conservation of mass forthe i-subregion a-phase is stated as:

d

dt�qi

aV ia � �

Xj 6�i

eija �2:2�

where the exchange terms eija express the mass source

terms of i-subregion a-phase from the j-subregion. Themass exchange terms are normalised with respect to Rand incorporate averaging in time. Their de®nition is:

eija �

1

2DtR

Zt�Dt

tÿDt

ZAij

a

nij � �q�wij ÿ v��dAds �2:3�

Conservation of momentum. The appropriate micro-scopic quantities for the thermodynamic property w, thenon-convective interaction i, the external supply of w, f ,and the internal production, G, which need to be substi-tuted into the generic balance Eq. (2.1), can be found inTable 2. First we employ f � r�/ÿ /0�. For reasons ofconvenience we also introduce a speci®c areal vector Aij

a :De®nition: We de®ne Aij

a as a time-averaged vector,representing the ¯uid exchange surface and normalisedwith respect to the REW's surface area projection R:

Aija �

1

2DtR

Zt�Dt

tÿDt

ZAij

a

nij dAds: �2:4�

Scalar multiplication of Aija with the unit vectors point-

ing, for example, along the axes of a Cartesian referencesystem, yield respective projections of area Aij

a onto theyz, xz and xy planes, respectively:

Aija;k � Aij

a � ek; k � x; y; z: �2:5�After introducing appropriate symbols for the momen-tum exchange terms, and employing the de®nitionEq. (2.4), we obtain for constant mass densities:

d

dt�qi

aviaV i

a �

�Xj 6�i

eija vi

a

264 � Tija ÿ

1

2D

Zt�Dt

tÿDt

ZAij

a

�/ÿ /0�nij dAds

375:�2:6�

The integral in the last term can be replaced by intro-ducing the average gravitational potential of the a-phasewith respect to a datum, calculated over Aij

a :

�/ija ÿ /i

0�Zt�Dt

tÿDt

ZAij

a

dAds �Zt�Dt

tÿDt

ZAij

a

�/ÿ /0�dA ds �2:7�

with /i0 a common reference potential for all phases of

the i-subregion. The operation yields the momentumbalance in the form:

d

dt�qi

aviaV i

a � �Xj 6�i

eija vi

a

� � Tija ÿ qi

a�/ija ÿ /i

0�Aija

�: �2:8�

The REW-scale momentum exchange terms Tija are

given by the expression:

Tija �

1

2DtR

Zt�Dt

tÿDt

ZAij

a

nij � �tÿ q�vÿ wij�~v�dAds; �2:9�

where the microscopic stress e�ects, attributable to thedeviations ~v of the REW-scale velocity from its spatialand temporal average, have been incorporated into themomentum exchange terms. Alternatively we can em-ploy the gravity vector as external supply of momentum,f � g, instead of the gravitational potential. Conse-quently we obtain an expression for the momentumbalance equivalent to Eq. (2.8):

d

dt�qi

aviaV i

a � � qiagi

aV ia �

Xj 6�i

eija vi

a

� � Tija

�: �2:10�

We emphasise that, while the use of Eq. (2.8) is neces-sary for the development of the constitutive relationship,Eq. (2.10) is employed as a governing equation, becausethe explicit consideration of the gravity vector will provemore useful, when projection of the equations along theaxes of a reference system is required, as will be shownin Section 6.

Conservation of energy. The balance equation fortotal energy includes the sum of kinetic, internal andpotential energies. The appropriate microscopic quan-tities w, i, f and G, which need to be substituted intoEq. (2.1), are reported in Table 2. For constant densitysystems we obtain:

d

dtEi

a

�� 1

2�vi

a�2�qi

aV ia

��Xj 6�i

eija Ei

a

�� 1

2�vi

a�2�

�Xj 6�i

Tija

� ÿ qia�/ij

a ÿ /i0�Aij

a

� � via �

Xj 6�i

Qija � qi

ahiaV i

a

�2:11�The REW-scale heat exchange terms Qij

a are de®ned as:

Qija �

1

2DtR

Zt�Dt

tÿDt

ZAij

a

nij � �q� t� q�/ÿ /0�I� � ~vÿ q�vÿ wij�

� � ~Eia � �~vi

a�2=2�dA ds: �2:12�We observe, that the surface integrals of the velocity,internal and kinetic energy ¯uctuations ~v, ~E

ia, and

�~via�2=2, can be envisaged as contributing to watershed-

scale heat exchanges. Therefore, these quantities havebeen incorporated into the heat exchange term Qij

a .


Balance of entropy. Finally, we obtain the balanceequations of entropy. After substituting the necessarymicroscopic values, by observing that the internal gen-eration of entropy, G, is non-zero, and introducingREW-scale exchange terms of entropy, we obtain:

d

dt�qi

agiaV i

a � �Xj6�i

eija gi

a

� � F ija

�� qiabi

aV ia � qi

aLiaV i

a ;

�2:13�where F ij

a is given in analogy to previous de®nitions forthe exchange terms:

F ija �

1

2DtR

Zt�Dt

tÿDt

ZAij

a

nij � �jÿ q�vÿ wij�~g�dAds �2:14�

where ~g are the ¯uctuations of the entropy from itsspace-time average over the i-subregion a-phase.

2.2. The second law of thermodynamics

The second law of thermodynamics states that thetotal production of entropy within a physical system hasto be always non-negative. In the present case, the sys-tem under consideration is taken to be the entire wa-tershed, which has been discretised into an ensemble ofM REWs. The decision to take the entire watershed asthe physical system, instead of a single REW or subre-gion, is dictated by the fact that the sign of the inter-REW and inter-subregion entropy exchange terms arenot known. According to the second law of thermody-namics, the total entropy production on a per-unit-areabasis, L, for the entire watershed composed of M REWs,obeys the following inequality:

L �XM

k�1

Xi

Xa

qiaLi

aV ia

" #k

P 0: �2:15�

The inequality (2.15) can be expressed in terms of thebalance equation of entropy Eq. (2.13) for the individ-ual phases and subregions by applying the chain rule ofdi�erentiation to the left-hand side term and making useof the conservation equation of mass Eq. (2.2):

L �XM

k�1

Xi

Xa

qiaV i

a

dgia

dt

" #k

ÿXM

k�1

Xi

Xa

Xj 6�i

F ija

" #k

ÿXM

k�1

Xi

Xa

qiabi

aV ia

" #k

P 0: �2:16�

2.3. Conditions of continuity (jump conditions)

In addition to the above balance laws, there are re-strictions imposed on the properties' exchange terms atthe boundaries, where di�erent phases, subregions orREWs come together. These restrictions simply express

the fact that the net transfer of a property betweenphases across an inter-phase boundary (e.g. water-gasinterface) needs to be zero. A similar restriction appliesto the net transfer of properties between subregions (e.g.across the water table, the channel bed or the saturatedland surface). This is equivalent to the assumption thatthe boundary surfaces are neither able to store mass,energy or entropy, nor to sustain stress. The total netexchange of a property across a boundary is zero. Theserestrictions are commonly known as jump conditions andare, as such, derived in the standard literature of con-tinuum mechanics (see e.g. Ref. [16]). The speci®c jumpconditions which supplement the balance equations formass, momentum, energy and entropy are stated spe-ci®cally in the next paragraphs.

Continuity of mass exchange. The jump condition formass expresses the fact that the net transfer of materialfrom the i and the j subregions towards the interface Aij

a

must equal zero:

eija � eji

a � 0: �2:17�Continuity of momentum exchange. The sum of forces

exchanged across the interface need to satisfy conditionsof continuity. These include apparent forces attributableto the mass exchange and the total stress force attrib-utable to viscous and pressure forces:

eija �vi

a ÿ vja� � Tij

a � Tjia ÿ qi

a�/ija

� ÿ /i0�

ÿ qja�/ji

a ÿ /j0��Aij

a � 0;�2:18�

where we have exploited the fact that Aija � ÿAji

a . Wealso note that in the case of interfaces within the same¯uid, the densities are equal and, therefore:

/ija

� ÿ /i0

�ÿ /jia

� ÿ /j0

� � 0 �2:19�yielding the jump conditions in the form derived inReggiani et al. [33]. For reasons of convenience in themanipulations of the entropy inequality in Section 4, thegravitational potentials are retained in the jump condi-tions.

Continuity of energy exchange. The continuity of ex-change of energy across the Aij

a interface requires thatthe transfer of internal and kinetic energies due to massexchange, and the work of the REW-scale viscous andpressure forces, obeys the following expression:

eija Ei

a

�ÿ Ej

a �1

2�vi

a�2h

ÿ �vja�2i�� Tij

a � via � Tji

a � vja

ÿ qia�/ij

a

� ÿ /i0�vi

a ÿ qja�/ji

a ÿ /j0�vj

a

� � Aija � Qij

a � Qjia � 0:

�2:20�Continuity of entropy exchange. The continuity of

exchange of entropy across the interface requires satis-faction of the condition:

eija �gi

a ÿ gja� � F ij

a � F jia P 0; �2:21�

where the inequality sign accommodates the possibilityof entropy production of the interface.


3. The global reference system

The conservation equation of momentum for thevarious subregions constitute vectorial equations, de-®ned in terms of components with respect to an ap-propriate reference system. In order to be able to employthe equation for the evaluation of the velocity, we haveto introduce a global reference system and e�ective di-rections of ¯ow, along which the vectorial equations canbe projected.

Saturated zone: In Ref. [33] we have assumed that theregional groundwater ¯ow can be exchanged betweenneighbouring REWs. A global reference system O needsto be introduced for the entire watershed, with respect towhich the whole ensemble of REWs can be positioned.The origin of the reference system can be placed forexample at the watershed outlet. We observe that otherlocations for the origin O can also be chosen, as long asthe reference elevation is positioned properly for theensemble of REWs. The global reference system O hastwo mutually perpendicular coordinates x and y lying inthe horizontal plane and a coordinate z directed verti-cally upwards. We introduce three unit vectors, ex, ey , ez,pointing along the three directions, respectively. Theforces acting on the water within the saturated zone willbe projected along the three directions by taking thescalar product between the conservation equation ofmomentum and the respective unit vector. The gravityvector has its only non-zero component along the ver-tical direction in the reference system O:

g � ÿgez: �3:1�Fig. 3 shows the global reference system O positioned atthe outlet of a watershed which has been discretised into13 REWs.

Unsaturated zone: The ¯ow in the unsaturated zonecan be considered as directed mainly along the verticaldirection from the soil surface downwards into the sat-urated zone or rising vertically upwards from the watertable due to capillary forces. The possibility of hori-zontal motion within the unsaturated zone is also con-sidered. We can project the balance equation formomentum in the unsaturated zone along the axes of theglobal reference system by taking the scalar productwith ex, ey and ez.

Saturated and concentrated overland: The ¯ow in theoverland ¯ow zones is occurring on a complex, curvili-near surface, for which only the direction of the e�ectivenormal to the surface with respect to the vertical or thehorizontal plane can be determined uniquely. There arein®nite possible directions for the choice of e�ectivetangents to the ¯ow surface. Fortunately, we know that,in the absence of micro-topographic e�ects on the landsurface, the ¯ow on the land surface crosses the contourlines perpendicularly, along the direction of steepestdescent. As a consequence the ¯ow can be assumed to be

e�ectively one-dimensional and to occur on a plane withan average inclination as depicted in Fig. 4. E�ects ofany departures from this assumption can be incorpo-rated into the constitutive parameterisation, but is leftfor future research.

The average angle of inclination c of the line ofsteepest descent with respect to the vertical direction orthe horizontal plane can be determined from topo-graphic data (e.g. digital elevation maps). In order toaccount for two separate e�ective slopes for the con-centrated and the saturated overland ¯ow zones we in-troduce the following two formulas:

co � cosÿ1�Ro=So� �3:2�and

cc � cosÿ1�Rc=Sc�: �3:3�The quantities So and Sc are the surface areas covered bythe saturated and the concentrated overland ¯ow sub-regions, while Ro and Rc are their respective projectionsonto the horizontal plane. Even though the direction of¯ow with respect to the global reference system cannotbe determined for overland ¯ow, we introduce ®ctitiousunit vectors which are tangent and normal to the in-clined ¯ow plane and labelled with no

t and non in the case

of the saturated overland ¯ow, as can be seen fromFig. 4. For the concentrated overland ¯ow the unitvectors nc

t and ncn are introduced.

Fig. 3. A real world watershed subdivided into 13 REWs and the

global reference system (Sabino Canyon, Santa Catalina Mountains,

SE-Arizona, Reproduced from K. Beven and M.J. Kirkby, Channel

Network Hydrology, Wiley.


Channel reach: Channel ¯ow occurs along a tortuouspath from the highest point at the REW inlet to thelowest point at the outlet. E�ects of the curvatures ormeandering of stream channels can sometimes have asigni®cant impact on the momentum exchanges andenergy dissipation, and would need to be factored in theconstitutive relations. For the present, however, we willassume that the channel ¯ow is directed along a straightline, whose angle with respect to the horizontal surfacecan be determined from topographic data by using thefollowing formula:

cr � sinÿ1�DH=Cr�; �3:4�where DH is the elevation drop of the channel bed be-tween inlet and outlet and Cr is the length of the curveforming the channel axis. Also in this case we introduceunit vectors tangent and normal to the e�ective directionof ¯ow, labelled with nr

t and nrn, respectively, as shown in

Fig. 4.

4. Constitutive theory

The system of equations in Section 2, derived for thedescription of thermodynamic processes in a REW,comprises in total 24 balance equations for the water,solid and gaseous phases in the unsaturated zone, thewater and the solid phases in the saturated zone andthe water phase in the two overland ¯ow zones and in

the channel. The total number of available equations isgiven by 8 mass balance equations, 8 (vectorial) mo-mentum balance equations and 8 energy balance equa-tions, respectively. In order to keep the theorydevelopment manageable, we adopt a number of sim-plifying assumptions which seem reasonable whenstudying runo� processes:

Assumption I. We consider the soil matrix as a rigidmedium. The respective velocities can, therefore, be setequal to zero:

vum � vs

m � 0: �4:1�In addition, the e�ects of dynamics of the gas phase

on the system are assumed to be negligible:

vug � 0: �4:2�

Assumption II. The system is unithermal, i.e., thetemperatures of all phases and subregions of the MREWs forming the watershed are equal to a commonreference temperature denoted by h. This assumption isvalid in the absence of (geo) thermal activities andlimited temperature excursions of the land surface. Theassumption is justi®ed whenever the focus of hydrologyis restricted to runo� processes, which is the case in thisstudy. It is clearly not justi®ed, if the focus is onevapotranspiration processes where thermal e�ects andwater vapour transport are critical. The unithermalassumption subsequently allows us to eliminate those

Fig. 4. E�ective directions of ¯ow and slope angles for the overland regions and the channel reach.


terms in the entropy inequality which account for theproduction of entropy due to heat exchanges amongsubregions and REWs.

Assumption III. We assume that we are dealing with asimple thermodynamic system. For such systems it isgenerally assumed that the external energy supply terms,b, are related only to external energy sources, h:

bia �

hia

h: �4:3�

We realise that the assumption of zero velocity of thegaseous phase is very restrictive and cannot be sustainedwhen studying evaporation. For the purposes pursuedwithin the framework of this paper, where the mainfocus is oriented towards the study of the water phasemotion, Assumption I allows signi®cant notationalsimpli®cations and simpli®es the derivation consider-ably. Inclusion of the motion of the gaseous phase willbe pursued in the future.

Assumption II can and should be relaxed if one isinterested in thermal processes and modelling of land-atmosphere interactions involving energy transfer.Assumption III is generally accepted in continuum me-chanics and is reported in the standard literature (seee.g. Ref. [16]).

We recall that there are still unknown quantities,represented by the exchange terms for mass, e, andmomentum, T, as well as the entropies g. The externalsupply term h in the balance equations for energy andthe gravity g are known functions.

Assumptions I and II allow us to eliminate the bal-ance equations for thermal energy and the balanceequations for mass and momentum for the gaseous andthe solid phases. The system is thus reduced to 5 massand 5 (vectorial) momentum balance equations (one foreach subregion). It is evident, that the unknown vari-ables and exchange terms involved in the equations ex-ceed by far the number of available equations, leading toan indeterminate system. The de®cit will have to beprovided for by constitutive relationships. For this pur-pose, a set of independent variables needs to be selected.The remaining unknowns are successively expressed asfunctions of the independent variables and are, there-fore, labelled as dependent variables.

For the system under study we expect, for example,the velocities of the ¯uid phases v, the saturation func-

tion su for the unsaturated zone, the volumina yuxu,ysxs, ycxc, yoxo and mrnr of the various subregions to beimportant descriptors of the system. We have chosen theproducts of two quantities as independent variablesbecause they always appear together in the equations.Finally, the mass densities of water also have to be in-cluded in the list of independent variables. The resultingcomplete list of these unknowns for all subregions issummarised in Table 3. We are thus confronted with asystem of 10 equations in 16 unknowns. To overcomethe de®cit of equations we will have to introduce con-stitutive relationships. The development of the consti-tutive equations will be pursued in a systematic fashion,in order to reduce arbitrariness in the possible choicesfor the parameterisations. For this purpose we will makeuse of the method of Coleman and Noll [5], based on theexploitation of the entropy inequality. This analysis willbe presented in the following sections.

4.1. Constitutive assumptions for the internal energies

The Coleman and Noll method involves postulationof the functional dependencies of the phase internal en-ergies. We adopt the fundamental approach pioneered byCallen [4] as a guideline for this development. Accordingto Callen, the extensive internal energies E for the ®vesubregions and respective phases are dependent on theirextensive entropy S, volume V and total mass M:

E � E�S;V;M�: �4:4�The Euler forms of the internal energies (see Ref. [4]) areobtained by taking a ®rst order Taylor series expansionof Eq. (4.4):

E � hSÿ pV� lM; �4:5�where h is the temperature, p is the pressure of the phaseand l is the chemical potential. The extensive internalenergy needs to be converted into a corresponding spe-ci®c internal energy, such as energy per unit mass orvolume. We select energy per unit volume as the variableof choice. The expression (4.5) is subsequently dividedby the volume:

E � hgÿ p � lq; �4:6�where E and g are the internal energy and the entropyper unit volume. With particular reference to the i-subregion a-phase, we speci®cally obtain:

Table 3

List of independent variables

Variable Nomenclature Number of unknowns

Water saturation su 1

Volumina yuxu; ysxs; ycxc; yoxo;mrnr 1, 1, 1, 1, 1

Water phase densities qu;qs;qc;qo;qr 1, 1, 1, 1, 1

Water phase velocity (vectorial) vu; vs; vc; vo; vr 1, 1, 1, 1, 1

Total 16


Ei

a � hgia ÿ pi

a � liaq

ia: �4:7�

Subsequently, we di�erentiate the internal energyEq. (4.7) with respect to time and rearrange:

dgia

dt� 1

hdE

i

a

dtÿ gi

a

hdhdt� 1

hdpi

a

dtÿ li

a

hdqi

a

dtÿ qi

a

hdli

a

dt: �4:8�

The Gibbs±Duhem equality, derived from the ®rst lawof thermodynamics [4], provides the information that:

giadhÿ dpi

a � qdlia � 0 �4:9�

so that Eq. (4.8), after multiplication by the volume V ia ,

assumes the form:

V ia

dgia

dt� V i

a

hdE

i

a

dtÿ V i

alia

hdqi

a

dt: �4:10�

Substitution of Eq. (4.10) into equation Eq. (2.16),where the entropy has been previously converted on aper-unit-volume basis, yields:

L �XM

k�1

Xi

Xa

V ia

1

hdE

i

a

dt

"ÿ li

a

hdqi

a

dt

!#k

� gia

dV ia

dt

ÿXM

k�1

Xi

Xa

Xj 6�i

F ija

" #k

ÿXM

k�1

Xi

Xa

qiabi

aV ia

" #k

P 0:

�4:11�We employ the conservation equation of energyEq. (2.11), together with the balance equation of massEq. (2.2) and the jump conditions for mass, momentumenergy and entropy Eqs. (2.17)±(2.21) to eliminate therate of change of internal energy and the exchange termfor entropy. This process is signi®cantly simpli®ed bymaking the following assumption.

Assumption IV. The phases are incompressible:

qia � constant:

We keep also in mind, that from Eq. (4.7) the pressurecan be expressed in terms of the temperature and thechemical potential:

ÿpia�h; li

a� � Ei

a ÿ hgia ÿ qi

alia: �4:12�

The outlined manipulations yield, with the use of As-sumptions I±IV, the ®nal expression:

L �XN

k�1

Xi

Xa

Xj6�i

1

hlj;i

a

�"� /j;i

a

�eij

a

#k

�XN

k�1

Xi

Xa

Xj 6�i

1

h

�"ÿ Tij

a � qia�/ij

a ÿ /0�Aija ÿ

1

2eij

a via

�� vi

a

#k

�XN

k�1

Xi

Xa

1

hqi

a�/ia

�"ÿ /0� � pi

a

�_V ia

#k

P 0; �4:13�

where /0 has been selected as the common referencepotential for all phases, subregions and REWs consti-tuting the watershed, and /i

a ÿ /0 is the gravitational

potential for the i-subregion a-phase, calculated withrespect to its centre of mass:

�/ia ÿ /0�qi

a

ZV ia

dV �ZV ia

q�/ÿ /0�dV : �4:14�

We observe that, while the microscopic function /ÿ /0

is independent of time, /ia ÿ /0 is a function of time.

4.2. The entropy inequality at equilibrium

According to the second law of thermodynamics, theentropy production of the entire system is always non-negative and will be zero only at thermodynamic equi-librium. To extract more information from the entropyinequality Eq. (4.13), the system will be analysed byimposing equilibrium. For the system of phases, subre-gions and REWs considered here, a situation of ther-modynamic equilibrium can be de®ned as thecon®guration where there is absence of motion and themass exchange terms between phases, subregions andREWs are zero. This can be expressed in quantitativeterms by stating that the following set of variables inEq. (4.13) are zero for the kth REW:

�zl�k � �via;

_V ia ; e

ija �k � 0: �4:15�

In addition, at equilibrium, all the temperatures of thedi�erent subregions and the surrounding environment(i.e. atmosphere, underlying strata) are at an equilibriumtemperature h, and none of the subregions is subject toexpansion or contraction in terms of respective volu-mina occupied by the subregions and phases. Further-more, the velocities of the water and the gaseous phasesare zero. The above de®ned set �zl�k identi®es a variablespace, wherein the entropy production L of the entirewatershed is de®ned. The situation of thermodynamicequilibrium is equivalent to L being at its absoluteminimum. At the origin of the variable space, where all�zl�k are zero, we have that L � 0. The necessary andsu�cient conditions to assure that L is at an absoluteminimum are that:

oLo�zk�k

� �e

� 0 �4:16�

and that the functional is concave around the origin,which requires the second derivative

o2Lo�zl�ko�zk�k

� �e

�� 4:17�

to be positive semi-de®nite. To further exploit the en-tropy inequality, an equilibrium situation needs to bede®ned for every subregion. These de®nitions arestrongly dependent on the hydrologic situations underconsideration and are presented in Section 5. Subse-quently, the inequality (4.13) will be di�erentiated withrespect to the set of variables Eq. (4.15) and condition


(4.16) needs to imposed. The result of this operationyields a series of equilibrium conditions. From the ®rstline of Eq. (4.13) we obtain that

lia � /i

a � lja � /j

a; �4:18�while the di�erentiation of the second line with respectto vi

a yields:

ÿTija je � qi

a�/ija ÿ /0�Aij

a � 0: �4:19�From the last term of Eq. (4.13) we obtain the equilib-rium condition

pia � qi

a�/ia ÿ /0� � 0: �4:20�

Combination of Eqs. (4.19) and (4.20) ®nally leads tothe equilibrium expression for the momentum exchangeterm:

Tija je � �ÿpi

a � qia�/ij

a ÿ /ia��Aij

a : �4:21�Eq. (4.21) needs to be projected along the axes of thereference system O introduced in Section 3, to evaluatethe pressure forces acting on the various phases. Theexpression can be used for two di�erent purposes: ®rst,if the pressure is known, Eq. (4.21) can be employed forthe evaluation of Tij

a je, second, if Tija je is known,

Eq. (4.21) will prove useful for the evaluation of pia.

4.3. Non-equilibrium parameterisation of the momentumexchange terms

Under non-equilibrium conditions, viscous forcesappear next to the pressure forces due to the motion ofthe ¯uid and the resulting frictional resistance. Conse-quently, we propose to add a non-equilibrium compo-nent to the sum of all equilibrium forces acting on thephases, which becomes zero at equilibrium:Xj 6�i

Tija �

Xj 6�i

Tija je � �si

a; �4:22�

where Tija je are given by Eq. (4.21) and �si

a is the non-equilibrium component of the forces. The non-equilib-rium part can be expanded as a ®rst or second orderfunction of the velocity, depending on which type of¯ow we are describing. This procedure is based on aTaylor series expansion. First-order approximations aresuitable for conditions of slow ¯ow, where second orderterms can be considered unimportant. This is especiallythe case for the ¯ow in the unsaturated and saturatedzones. The linearised non-equilibrium term assumes thefollowing form:

�sia � ÿRi

a � via; �4:23�

where Rj is a tensor, which is a function of the remainingindependent variables and parameters of the system. Forthe ¯ow occurring on the land surface and in thechannel, experimental observations suggest that thefriction forces depend on the square of the velocity. Thisfact is evident from formulations such as Chezy's and

Manning's law, which serve as parameterisations for thebottom friction at the local scale in situations of over-land and channel ¯ow. As a result, a second-order ap-proximation in terms of the velocity is sought in thefollowing fashion:

�sia � ÿRi

a � via ÿ jvi

aj �Uia � vi

a; �4:24�where, once again, Ri

a and Uia are tensors which depend

on some of the remaining variables and parameters. Weobserve the necessity to take the absolute value of thesecond order velocity to preserve the sign of the ¯owresistance force, which is always directed opposite to the¯ow.

5. Closure of the equations

The balance equations for mass, momentum, energyand entropy for all ®ve subregions of the watershed havebeen rigorously derived by Reggiani et al. [33]. Thanksto Assumption I we can generally omit the subscripts a,which indicate di�erent phases, as we are dealing withwater in all subregions as the only mobile phase. Thebalance equations and relative exchange terms for massand momentum, reported throughout the subsequentsections, refer, therefore, to the water phase only. Sub-sequently, we need to ®nd expressions for the momen-tum exchange terms in the balance of forces. Second,parameterisations for the mass exchange terms need tobe proposed. These two issues are handled separately inthe following sections.

5.1. Closure of the momentum balance equations

Unsaturated zone: The speci®c balance equation ofmomentum for the unsaturated zone water phase, statedin the form Eq. (2.10), is:

�q�yusuxu� d

dtvu ÿ q�yusuguxu

� TuAje � Tusje � Tucje � Tuwgje � Tu

wmje ÿ Ru � vu �5:1�where yu is the average thickness of the unsaturatedzone, � is the porosity of the soil matrix, su is the waterphase saturation and xu is the area fraction covered bythe unsaturated zone. The right-hand side terms are theequilibrium components of the following forces: TuA isthe force exerted on the prismatic mantle surface, Tuc isthe force acting on the unsaturated zone water phaseacross the land surface, while Tu

wg and Tuwm are the forces

exchanged between water and gas and water and soilmatrix, respectively. All quantities have been accuratelyde®ned by Reggiani et al. [33]. Furthermore, we haveexpanded the non-equilibrium components of theseforces as a function, which is linear in the velocity, assuggested in Section 4.3.


To obtain expressions for the equilibrium forces, anappropriate condition of equilibrium for the unsaturat-ed zone needs to be de®ned. We describe equilibrium asthe situation where the forces acting on the water phasewithin the soil pores (gravity, capillary forces) and theforces acting between REWs across the lateral mantlesurface are balanced. In this case the average waterphase velocity is zero and the mass exchanges across thewater table, the mantle and the land surface are all zero.The same is valid for the phase changes between waterand vapour within the soil pores. By applying theequilibrium condition (4.21), the speci®c expressions forthe force terms at equilibrium are obtained. First, wenote the equilibrium force acting on the water phasethrough interaction with the soil matrix and the gaseousphase is equally zero for symmetry reasons:

Tuwgje � Tu

wmje � 0: �5:2�Next, the force acting on the prismatic mantle surface isgiven by the expression:

TuAje � �ÿpu � q�/uAÿ /u��AuA: �5:3�We recall from Reggiani et al. [33] that the mantle sur-face is composed of a series of segments which form theboundary between the REW under consideration andneighbouring REWs or the external boundary of thewatershed. Therefore, the total force TuAje needs to beseparated into the various components acting on therespective mantle segments:

TuAje �X

l

TuAl je � TuA

extje: �5:4�

The force Tucje, acting on the land surface, can be de-rived from Eq. (4.21) in complete analogy to Eq. (5.3).The total force Tusje, acting on the water phase along thephreatic surface is e�ectively zero, as the pressure isatmospheric in these locations. Thus we impose thecondition:

�ÿpu � q�/us ÿ /u��Aus � 0 �5:5�from which we obtain an expression, which can be em-ployed for the evaluation of the equilibrium pressure inEq. (5.3) and its counterpart for Tucje:pu � q�/us ÿ /u�: �5:6�After substituting the equilibrium forces into Eq. (5.1),the equation needs to be projected along the axes of thereference system O introduced in Section 3 to evaluatethe actual components of the equilibrium forces. Thistask will be pursued in Section 6.

Saturated zone: The balance equation of momentumfor the saturated zone is:

�q�ysxs� d

dtvs ÿ q�gsysxs

� TsAje � Ts botje � Tsuje � Tsoje � Tsrje � Tswmje ÿ Rs � vs;

�5:7�

where ys is the average thickness of the saturated zoneand xs is the area fraction covered by the aquifer. Theright-hand side terms are the respective equilibriumcomponents of the REW-scale force exerted on theprismatic mantle surface, TsA, the force acting on thebottom of the aquifer, Tsbot, on the water table, Tsu, onthe seepage face, Tso, on the channel bed, Tsr, and onthe soil matrix within the porous medium, Ts

wm. Alsoin this case the non-equilibrium component of theforces has been expressed as a linear function of thevelocity.

The condition for mechanical equilibrium within thesaturated zone is de®ned as the situation, where allforces acting on the water phase are balanced, and themass exchanges across the water table and the prismaticmantle surface are zero. The mass exchanged across thebed surface of the channel is also zero because the dif-ference in hydraulic potentials between the channel andthe saturated zone is zero at equilibrium. With theseconsiderations in mind, we ®rst ®nd that the equilibriumforce exerted by the water on the soil matrix is zero forsymmetry reasons:

Tswmje � 0: �5:8�

In analogy to previous cases, we obtain the equilibriumforce acting on the mantle through application ofEq. (4.21):

TsAje � �ÿps � q�/sA ÿ /s��AsA: �5:9�This force needs, once again, to be separated into thecomponents acting on the segments of the various seg-ments of the mantle, which separate the aquifer form theaquifers of neighbouring watersheds or are part of theexternal watershed boundary. Similar expressions forthe forces Tsoje and Tsrje, acting on the seepage face andthe channel bed, respectively, can be obtained fromEq. (4.21). Finally, from the momentum balanceEq. (5.7) we obtain that the total equilibrium forceTs botje, acting on the bottom of the aquifer, must bal-ance the weight of the water contained within the satu-rated zone:

Ts botje � ÿqgsys�xs � �ÿps � q�/s bot ÿ /s��As bot:

�5:10�This equation can be employed to evaluate the averagewater pressure ps, once it has been projected along thevertical direction through scalar multiplication with theunit vector ez.

Concentrated overland ¯ow: The balance equation ofmomentum for the concentrated overland ¯ow zone (inthe form Eq. (2.10)) is:

�qycxc� d

dtvc ÿ qycgcxc

� Tc topje � Tcuje � Tcoje ÿ Rc � vc ÿ jvcj �Uc � vc;

�5:11�


where yc is the average thickness of the ¯ow sheet and xc

is the area fraction covered by the concentrated over-land ¯ow. The terms on the right-hand side are theequilibrium components of the forces exchanged withthe atmosphere, Tctop, with the unsaturated zone, Tcu

and with the saturated overland ¯ow, Tco, respectively.The non-equilibrium component has been expanded as asecond order function of the velocity. For the concen-trated overland ¯ow, we de®ne equilibrium as the situ-ation where there is no ¯ow. Due to the fact that thewater is ¯owing along a slope, no-¯ow conditions canonly be achieved under complete absence of water al-together (dry land surface). In this case vc, yc and theaverage pressure pc are zero. Consequently Eq. (4.21)yields:

Tcoje � Tcuje � 0: �5:12�After we neglect the ®rst-order term in Eq. (5.11) andsubstitute Eq. (5.12), we obtain:

�qycxc� d

dtvc ÿ qycgcxc � ÿjvcj �Uc � vc �5:13�

Saturated overland ¯ow: In complete analogy to theprevious case, we de®ne equilibrium as the situation ofno ¯ow, which eventuates in the absence of water on thesaturated areas. This de®nition yields zero momentumexchange terms at equilibrium and the conservation ofmomentum reduces to:

�qyoxo� d

dtvo ÿ qyogoxo � ÿjvoj �Uo � vo: �5:14�

We observe, that in di�erent hydrologic situations, adi�erent de®nition of equilibrium could have beenadopted. In the case of wetlands, where the saturatedoverland ¯ow zone consists of stagnant water with near-horizontal free surface covering the soil, equilibrium isequivalent to zero ¯ow velocity vo and non-zero waterdepth yo. In this case non-zero equilibrium forces andpressure po would have been obtained.

Channel reach: The speci®c conservation equation formomentum for a single reach can be stated in the form(for reference see Eq. (2.10)):

�qmrnr� d

dtvr ÿ qmrgrnr

� TrAje � Tr topje � Trsje � Troje ÿ vr �Ur � vr

�5:15�

where mr is the average cross sectional area of the reachand nr is the length of the channel axis on a per-unit-areabasis. The forces on the right-hand side are exchangedon the channel end sections, TrA, on the channel freesurface with the atmosphere, Tr top, with the channelbed, Trs, and with the saturated overland ¯ow on thechannel edges, Tro. Also here the ®rst order term of thefrictional force has been omitted.

Equilibrium in a channel can be de®ned by assuminga near-horizontal free surface within the reach. Underthese circumstances the ¯ow velocity vr is zero. A dif-

ferent equilibrium condition, such as a dry channel,could be imposed, if the situation would suggest it (e.g.case of a steep channel). From the equilibrium expres-sion for the momentum exchange terms Eq. (4.21) weget the expression:

TrAje � �ÿpr � q�/rA ÿ /r��ArA �5:16�for the total force acting on the end sections of thechannel reach. We remind the reader (for reference seeReggiani et al. [33]) that the surface ArA represented bythe vector ArA constitutes a single cross section at theoutlet, if the REW under consideration is relative to a®rst order stream or includes two inlet sections (for eachof the two reaches converging at the inlet from up-stream) and an outlet section in the case of higher orderstreams:

TrAje �X

l

TrAl je � TrA

extje �5:17�

TrAextje is non-zero only for the reach, whose end section

coincides with the watershed outlet. In analogy toEq. (5.16) we obtain an expression for Trsje fromEq. (4.21). Finally, we recall that the atmosphericpressure force acting on the channel surface at equilib-rium, Tr topje, is zero, yielding an equation which isuseful for the evaluation of the average pressure pr

within the reach:

pr � q�/r top ÿ /r�: �5:18�

5.2. Linearisation of the mass exchange terms

The mass exchange terms are unknown quantities ofthe problem. From the entropy inequality (4.13), a li-nearisation of the mass exchange terms as functions ofdi�erences in chemical potentials, gravitational poten-tials and average velocities within the adjacent subre-gions is suggested:

eija �Aij lj

a

� ÿ lia � /j

a ÿ /ia

�ÿBij 1

2�vi � vj� � Aij

a

�5:21�where the areal vector Aij

a represents the ¯uid fraction ofthe interface and is de®ned through Eq. (2.4). The li-nearisation coe�cients Aij and Bij in Eq. (5.21) arefunctions of the remaining independent variables andsystem parameters and are both positive. We point out,that the chemical potential can be expressed in terms ofthe internal energy, temperature and entropy (for ref-erence see Eq. (4.7)):

lia � Ei

a ÿ hgia �

pia

qia

�5:20�

For situations of unithermal, slow ¯ow, where the dif-ferences in internal energy and entropy are negligible,the mass exchange term can be linearised in terms ofdi�erence in hydraulic potentials:


eija �Aij pj

a

� ÿ pia � qj

a/ja ÿ qi

a/ia

�ÿBij 1

2�vi � vj� � Aij

a

�5:21�Experimental e�ort is required in order to establish thevarious functional dependencies of the coe�cients. Herewe discuss the proposed parameterisations for all massexchange terms on a case-by-case basis.

Unsaturated zone ± concentrated overland ¯ow (euc):The mass exchange term expressing the in®ltration ofwater across the soil surface into the unsaturated zone islinearised in terms of the di�erence in hydraulic poten-tials between the concentrated overland ¯ow (coveringthe unsaturated portion of the REW land surface) andthe unsaturated zone. The average water pressure pc inthe concentrated overland ¯ow subregion is comparablewith atmospheric pressure, whereas the average waterpressure pu in the pores of the unsaturated zone is lowerthan the atmospheric pressure:

euc � ÿecu �Auc pc� ÿ pu � q�/c ÿ /u�� 5:22�The term euc acts as source for the unsaturated zone andas sink term for the concentrated overland ¯ow. Con-sequently the constant Auc needs to be always positive.

Unsaturated zone ± saturated zone (eus): The mass ¯uxacross the water table is linearised here in terms of thevelocities vu and vs of the unsaturated and the saturatedzones:

eus � ÿesu � ÿBus 1

2�vu � vs� � Aus �5:23�

where Aus is the area vector representing the water tableas de®ned by Eq. (2.4) (the subscript w indicating thewater fraction of the interface has been omitted). In thisfashion, the mass exchange term can change sign ac-cording to the direction of the average velocity (upwardsand downwards) and can switch between recharge of thesaturated zone and capillary rise from the water tabletowards the unsaturated zone. The linearisation pa-rameter Bus is thus required to be positive.

Unsaturated zone: water±vapour phase change (euwg):

Within the unsaturated zone water can change phase,turn into vapour, and leave the system through the soilsurface after exhaustion of the concentrated overland¯ow, leading to soil water evaporation. The transitionfrom water to water vapour is dependent on the di�er-ence in chemical potentials between the two phases,while their di�erence in gravitational potentials is zerowithin the same subregion:

euwg � ÿeu

gw �Auwg�lu

g ÿ luw�: �5:24�

The chemical potentials of the two phases are functionsthe respective pressures of the water and the gas phases,the vapour mass fraction in the gas phase and thetemperature. The analysis of this term is beyond thescope of this work and will be pursued in the future. Formore in-depth knowledge about this subject the reader is

referred to speci®c literature in the ®eld of soil physics,e.g. Nielsen et al. [31] or Hillel [25].

Unsaturated and saturated zones, mass exchangeacross the mantle (euA, esA): With respect to the lineari-sation of the mass exchange across the mantle surface Asurrounding the REW laterally, a series of preliminaryconsiderations are necessary. We recall that the REW issurrounded by a number Nk of neighbouring REWs.The REW under consideration exchanges mass with thesurrounding REWs across respective segments Al of themantle surface and across the external boundary of thewatershed. The mantle surface A can, therefore, bewritten as a sum of segments:

A �X

l

Al � Aext �5:25�

where Al forms the boundary with the lth neighbouringREW and Aext coincides with the external watershedboundary. For example, REW 1 in Fig. 3 has a num-ber of Nk � 3 neighbouring REWs and one mantlesegment in common with the external watershedboundary. The index l assumes the values 2, 3, and 4.As explained by Reggiani et al. [33], the latter term isnon-zero only for REWs which have one or moremantle segments in common with the watershedboundary. A typical contact zone between two neigh-bouring REWs through the mantle is depicted sche-matically shown in Fig. 5. As a result, the total mass¯ux eiA, i � u; s, can be written as the sum of the re-spective ¯ux components relative to the various mantlesegments:

eiA �X

l

eiAl � eiA

ext i � u; s �5:26�

We propose a constitutive parameterisation for eiA, byassuming a dependence of the exchange term on the thearea Al of the prismatic mantle segment and the veloc-ities in the REW under consideration and in the lthneighbouring REW, vi and vijl, respectively:

eiAl � ÿBiA

l

1

2�vi � vijl� � Al �5:27�

where the vector Al is de®ned through Eq. (2.4) andexpresses the ¯uid fraction of the lth mantle segment. Ina similar fashion, the mass exchange terms for the sub-surface zones of the kth REW across the external wa-tershed boundary Aext can be expressed as:

eiAext � ÿBiA

ext

1

2�vi � vijext� � Aext �5:28�

where vijext is depending on boundary conditions.Saturated zone ± channel (esr): The saturated zone can

be recharged from the channel through seepage acrossthe bed surface or can drain water towards the channel.Recharge/drainage should be possible even when theaverage velocity within the saturated zone is zero.Therefore, we propose a linearisation, which is depen-dent only on the di�erence in hydraulic potentials


between the channel and the saturated zone, and isindependent of the velocity:

esr � ÿers �Ars pr� ÿ ps � q�/r ÿ /s�� 5:29�Saturated zone ± saturated overland ¯ow (seepage)

(eso): As in the case of exchange with the channel, theseepage from the saturated zone also can eventuate inthe absence of an average motion within the aquifer.Hence, the seepage out¯ow is parameterised in analogyto Eq. (5.29) in terms of the hydraulic potentials di�er-ence between the two subregions:

eso � ÿeos �Ars po� ÿ ps � q�/o ÿ /s�� 5:30�Concentrated overland ¯ow ± saturated overland ¯ow

(eco): The mass exchange between the regions of con-centrated and saturated overland ¯ow is occurring alongthe perimeter, circumscribing the saturated areas exter-nally. This perimeter is de®ned by the line of intersectionof the water table with the land surface. The cross sec-tional areas, across which the ¯ow from uphill connectswith the saturated overland ¯ow, are denoted with Aoc,and are represented by an appropriate vector Aco. Themass exchange term between the two subregions isproposed as a linear function of the mean velocity�vc � vo�=2:

eco � ÿeoc � ÿBco 1

2�vc � vo� � Aco �5:31�

where Bco is a positive coe�cient.Saturated overland ¯ow ± channel (esr): The lateral

in¯ow from the areas of saturated overland ¯ow into thechannel occurs along the channel edge. the ¯ow cross

section in these zones is denoted with Aor and is repre-sented by a respective areal vector Aor de®ned byEq. (2.4). The mass exchange can be assumed as linearin average velocity vo (the velocity in the channel isdirected parallel to the edge and does not contributeto esr):

eor � ÿero � ÿBorvo � Aor �5:32�where Bor is a positive coe�cient.

Channel in¯ow and out¯ow (erA): The channel networkconstitutes a bifurcating tree. The total mass exchangeerA of the channel reach across the mantle of the kthREW can be separated into the two counterparts at-tributable to the REWs converging at the inlet sectionand the mass exchange with the REW further down-stream. In the case of REWs associated with ®rst orderstreams, the reach has only one out¯ow. This can beseen best in Fig. 3. For example, the channel reach ofREW 7 communicates with the reaches of REW 5 and 6at the inlet section and with REW 9 at the outlet(Nk � 3). REW 1 has only one out¯ow towards REW 4(Nk � 1). As a result, we write the total mass exchange ina general fashion as the sum of the following constituentparts:

erA �X

l

erAl � erA

ext �5:33�

where the second term on the right-hand side is non-zero only for the REW located at the outlet (e.g. REW13 in Fig. 3). The sum extends over a single neigh-bouring REW in the case of ®rst order streams andover three REWs in the case of higher order streams.

Fig. 5. Detailed view of the contact zone between two adjacent REWs.


The proposed linearisation of the mass exchange termsis expressed through the cross sectional area vector ArA

land the mean velocity �vr � vrjl�=2 between the reach ofthe kth REW and the reach of the lth neighbouringREW:

erAl � ÿBrA

l

1

2�vr � vrjl� � ArA

l �5:34�The coe�cients BrA

l are positive. A similar equation maybe given for erA

ext:

erAext � ÿBrA

ext

1

2�vr � vrjext� � ArA

ext �5:35�

where vrjext is assumed to be known. For example, in thecase of a river ¯owing into a lake vrjext may be set equalto zero.

Rainfall or evaporation (ec top, eo top, er top): The massexchange between the concentrated and the saturatedoverland ¯ow and the atmosphere can be expressed aslinear functions of the respective area fraction of thesubregion and the rate J of mass input (rainfall inten-sity) or extraction (evaporation rate):

ei top � xiJ i � c; o �5:36�The mass exchange with the atmosphere on the channelfree surface can be expressed as a linear function of themass input or extraction J and the area of the channelfree surface Ar top � wrnr, where wr is the averagechannel top width, to be de®ned in Section 7:

er top � wrnrJ : �5:37�

6. Parameterised balance equation

In the previous section we have proposed possibleparameterisations of the mass and momentum exchangeterms. Here we project the momentum balance equa-tions along the reference system introduced in Section 3,and substitute the respective exchange terms for massand momentum into the respective conservation equa-tions. The mass balance equations have been derived byReggiani et al. [33] and are written in the general formEq. (2.2). For reason of simplicity of the ®nal set ofequations, we state a series of assumptions regarding thegeometry of the REW:

Assumption V. The water table within the REW is near-horizontal and the slope of the land surface is small. Asa result we obtain that the horizontal components of thevectors Aij de®ned through Eq. (2.4) for the unsaturatedland surface, Auc, the seepage face, Aso, and the watertable, Aus, are negligible. The same is valid for the cannelbed surface, Asr:

Auc � ek � Aso � ek � Aus � ek � Asr � ek � 0; k � x; y

�6:1�

Assumption VI. The bottom boundary of the aquifer isimpermeable. As a result the vertical ¯ow across thesaturated zone is zero:

vs � ez � 0 �6:2�

Assumption VII. The tangential component of the vectorrelative to the channel bed area, Ars, is negligible:

Ars � nrt � 0 �6:3�

We emphasise that these assumptions can be relaxed ifrequired by the particular circumstances.

6.1. Unsaturated zone

Balance of mass: In view of Assumption V and fromde®nition Eq. (2.4) we ®nd that the ¯uid component ofthe horizontal exchange surface is:

Auc � ez � �xuR �6:4�Subsequently, we introduce the linearised mass ex-change terms Eqs. (5.22)±(5.24) and Eq. (5.27) into theunsaturated zone mass balance in the form Eq. (2.2) andobtain:

�6:5�where the signs in the last term are positive or negativeaccording to the orientation of AuA

l with respect to theglobal reference system O. The mass exchange euA

ext acrossthe mantle segment overlapping with the watershedboundary needs to be imposed according to the boun-dary conditions, e.g. by Eq. (5.28) or by zero-¯uxboundary conditions, euA

ext � 0.Balance of momentum: Considering that the ¯ow in

the unsaturated zone is directed mainly along the ver-tical, we ®rst project the conservation equation of mo-mentum (5.1) through scalar multiplication with the unitvector ez. We also note that the z-coordinate is positiveupward so that gu � ez � ÿg. Thus we obtain in view ofEq. (6.4) the linearised form of the momentum balancealong ez:

�6:6�where the inertial term has been considered negligibleand the resistivity is isotropic. Along the horizontal di-rections pointed to by ex and ey , the components of the


gravity are zero. In view of Assumption V, the mo-mentum balance along ex and ey becomes:

�6:7�where the signs are either positive or negative accordingto the orientation of AuA

l and AuAext with respect to the

reference system O. We note that, in the case of fast¯ows, where the resistance no longer varies linearly withvelocity, second or higher order approximations for theresistance term can be sought. The force TuA

ext is non-zeroonly for REWs with a mantle segment in common withthe external watershed boundary and have to be im-posed according to the actual boundary conditions (e.g.zero force, constant or time-varying pressure forceacting on the boundary). The momentum balanceequations given here are equivalent to a generalisedDarcy's law for the unsaturated zone at the scale of theREW.

6.2. Saturated zone

Balance of mass: The ¯ow in the saturated zone isassumed to occur only in a horizontal plane parallel tothe xÿ y plane of the global reference system O (SeeAssumption VI). We introduce the linearised mass ex-change terms Eqs. (5.23), (5.27) and (5.29) andEq. (5.30) into the equation of conservation of mass forthe saturated zone (stated in the form Eq. (2.2). Thesubstitution yields:

�6:8�where esA

ext is given according to the boundary conditions.The signs in the last term on the right-hand side becomepositive or negative according to the orientation of thesurfaces with respect to the global reference system O. Itcan be no ¯ux, or a speci®ed mass in¯ow or out¯ow asgiven for example by Eq. (5.28). The last line in thisequation allows the integration of regional groundwater¯ow into the watershed equations.

Balance of momentum: The balance equation of mo-mentum is obtained from Eq. (5.25) through scalar

multiplication with the unit vectors ex and ey . RecallingAssumption V we obtain:

�6:9�This equation can be interpreted as a REW-scale Dar-cy's law for the saturated zone.

6.3. Concentrated overland ¯ow zone

Conservation of mass: We introduce the mass ex-change terms Eqs. (5.22) and (5.31) and the term ac-counting for rainfall input Eq. (5.36) into theappropriate equation of conservation of mass presentedby Reggiani et al. [33]:

�6:10�We note that the projection of the cross sectional areaAco has been approximated by the product:

Aco � Kco 1

2�yo � yc� �6:11�

where Kco is the length of the contour curve forming theperimeter of the saturated areas. The length of the curveis subject to variations due to possible expansions andcontractions of the seepage faces. A plausible geometricrelationship will be presented in Section 7.

Conservation of momentum: The ¯ow has been as-sumed to occur along an e�ective direction tangent to a¯ow plane with an inclination angle cc with respect tothe horizontal, as explained in Section 4. A unit vectornc

t , pointing along the e�ective direction, has been in-troduced, as shown in Fig. 4. We now project theequation of conservation of momentum derived fromEq. (5.11) by scalar multiplication with nc

t :

�6:12�

If we also neglect the inertial term, we obtain a REW-scale Chezy formula for the concentrated overland ¯ow:

qycxcg sincc � ÿU cvcjvcj: �6:13�


6.4. Saturated overland ¯ow zone

Conservation of mass: The conservation of mass forthe saturated overland ¯ow subregion is linearised byintroducing the parameterised mass exchange termsEqs. (5.30) and (5.32) and the term Eq. (5.36) account-ing for the rainfall input into the appropriate massbalance equation:

�6:14�where the area Aor has been approximated by theproduct:

Aor � Koryo �6:15�with Kor is the contour curve forming the edge of thechannel, to be addressed in Section 7.

Conservation of momentum: The momentum balanceequation Eq. (5.32) needs to be projected along thetangent to the respective ¯ow plane (see Fig. 4) throughscalar multiplication with no

t . The result is:

�qyoxo� d

dtvo ÿ qyoxog sinco � ÿU ovojvoj: �6:16�

Omission of the inertial term leads to a REW-scaleChezy-formula.

6.5. Channel reach

Conservation of mass: The conservation equation ofmass for the channel reach belonging to the kth REW isparameterised by introducing the linearised mass ex-change terms Eqs. (5.29) and (5.32) and Eq. (5.34) intothe appropriate mass balance equation for the reachpresented by Reggiani et al. [33]:

�6:17�where the sign of the second-last term is positive for inletsections (source of mass for the reach) and negative foroutlet sections (mass sink). The term erA

ext is non-zeroonly for the last REW whose outlet coincides with theoutlet of the entire watershed (e.g. REW 13 in Fig. 3)and is given by Eq. (5.35). Eq. (6.17) is in completeagreement with the mass conservation equation for abinary tree forming a channel network, as derived by

Gupta and Waymire [20]. In the present work we gofurther by formulating the corresponding momentumbalance equation for the network, as described below.

Conservation of momentum: The conservation equa-tion for momentum is obtained from Eq. (5.15) throughprojection of each reach along its e�ective tangent to thechannel bed nr

t , shown in Fig. 4. Consequently, weconsider Assumption VII and employ simply the symbolvr to denote average along-slope channel velocity:

�6:18�where the last term is non-zero only for the REW situ-ated at the watershed outlet and depends on the localboundary conditions.The sign of the second-last term isnegative for the inlet and positive for the outlet sectionsof the reaches. The angle dl is the local angle between thereach of the REW l and the reach of the REW underconsideration, as can be seen from Fig. 6. The angle dl

can be estimated from topographic data. In this way themomentum of the two upstream reaches converging atthe REW inlet is projected along the direction of thedownstream reach The two components of momentumnormal to the axis of the down-stream reach are as-sumed to cancel each other. We also observe, that theabove equation is equivalent to the Saint±Venantequation stated for a tree-like structure of zero-dimen-sional inter-connected buckets, where channel curvaturee�ects have been neglected. The pair of Eqs. (6.17) and

Fig. 6. Con¯uence angles of two reaches at the inlet of a REW.


(6.18) together govern the response of the channel net-work.

7. Discussion of the equation system

In the previous section we have obtained a system of13 non-linear, coupled balance equations for mass andmomentum for each subregion in the kth REW, whichrespect the jump conditions for mass and momentumacross the inter-phase, inter-subregion and inter-REWboundaries. The 19 unknowns of the system are:

Zk � �vux ; v

uy ; v

uz ; v

sx; v

sy ; v

o; vc; vr; su; yuxu; ysxs; ycxc; yoxo;

mrnr; pu; ps; pc; po; pr�k�7:1�

The 13 balance equations in Section 5 are supplementedwith Eqs. (5.6), (5.10) and (5.18) for the evaluation ofthe pressures. Based on the equilibrium assumptionsmade for the overland ¯ow zones (see Section 6) thepressures pc � 0 and po � 0. As a result, the number ofunknowns in Eq. (7.1) is reduced to 14 (one excess un-known with respect to the available equations). Never-theless, the balance equations allow us only to evaluatethe products of variables, such as xiyi or mrnr, while thearea fractions xi, the average channel cross-sectionalarea mr, the channel top width wr or the depth yr appearin the equations (or are needed indirectly for the eval-uation of the pressures) and are necessary for the eval-uation of the mass exchange terms. The same is valid forKoc and Kor. Consequently, 9 additional unknowns havebeen introduced. In total, there is a need of 10 additionalconstitutive relationships, to obtain a determinate sys-tem. These need to take explicitly into account the ge-ometry of the REW.

7.1. Constitutive relationships for geometric variables

To overcome the remaining de®ciency of equations,we introduce a set of 10 geometric relationships. The®rst relationship is based on the conservation of volumefor the entire subsurface region of the REW, includingthe saturated and the unsaturated zones. This relation-ship takes into account the fact that the total volume ofthe subsurface zone of REW (the saturated and theunsaturated zones) is constant. An increase or decreaseof the saturated zone due to ¯uctuations of the watertable results in an equal decrease or increase of thevolume of the unsaturated zone, respectively. Thisconsideration leads to a constitutive function which re-lates the rate of change of volume yuxu of the unsatu-rated zone to the rate of change of volume ysxs of thesaturated zone:

_�yuxu� � _�ysxs� � 0: �7:2�

In the parameterised equations, we ®nd that a know-ledge of the area fractions xj, the depths yj, thedrainage density nr or the cross sectional area mr isnecessary for the evaluation of the terms accounting forrainfall input and evaporation (eo top, ec top, er top), andthe mass and momentum exchange terms within theREW (ecu, eos, eus, eor, eoc, erA, esA, euA, eu

wg, �su, �ss, �so, �sc,�sr). To overcome this obstacle, we state another as-sumption:

Assumption VIII. The saturated zone underlies the wholeREW, and therefore, Rs � R. This implies that:

xs � 1: �7:3�For the area fraction of the saturated overland ¯ow xo,we postulate a dependence on the average depth of thesaturated zone ys and the rate of change of depth _ys:

_xo � _xo�ys; _ys�: �7:4�This functional dependence is related to the local to-pography of the REW. Such relationships can be de-rived by approaches similar to the topography-basedframework adopted by TOPMODEL of Beven andKirkby [1]. For example, the saturated area fraction ofthe catchment may be expressed as a function of thestatistical distribution of the topographic indexln�a�= tanb and the average depth of the water table,which is equivalent to the variable yu in the presentapproach. The area covered by the saturated overland¯ow (seepage faces) is complementary with the areacovered by the concentrated overland ¯ow zone, suchthat:

Ro � Rc � R �7:5�We observe that by assuming this relationship betweenarea projections, we have regarded the projection of thefree surface area Ar top of the channel as negligible. Theequality (7.4) yields, after division by R and di�erenti-ation with respect to time, another useful relationshipbetween the rate of change of area fractions:

_xo � _xc � 0 �7:6�Another relationship derives from the fact, that the areaprojection of the concentrated overland ¯ow region isoverlapping with the horizontal projection of the un-saturated zone, i.e. Ru � Rc. This leads to a relationshipbetween the two respective area fractions:

xu � xc �7:7�Finally, the drainage density nr can be assumed to be afunction of the average cross sectional area of thechannel mr and its rate of change:

_nr � _nr�mr; _mr� �7:8�We observe that this relationship is only valid for REWsrelated to ®rst order channels which are allowed to varytheir length through uphill expansion. If the REW isassociated with a higher order channel, the drainage


density remains constant and may be obtained fromtopographical data:

nr � known constant �7:9�The knowledge of the average channel top width, wr andthe average depth yr are convenient for the evaluation ofthe channel free surface area Artop � nrwr and thechannel pressure pr de®ned in Eq. (5.18). The averagewidth wr (or alternatively the average depth yr) can beobtained according to Leopold and Maddock [29]through a power law relationship in terms of the dis-charge Dr given the product of mr and the velocity vr (at-a-station hydraulic geometry):

yr � a�Dr�b � a�mrvr�b �7:10�wr � c�Dr�d � c�mrvr�d �7:11�The coe�cients a; c and the exponents b; d need to beevaluated from ®eld data. The length Kco of the inter-section curve between the water table and the landsurface changes with expansion or contraction of thearea of the seepage faces:

Kco � Kco�xo�: �7:12�This function could be obtained from topographicmaps. The length of the channel edge Kor is a function ofthe top width of the channel and the total length:

Kor � Kor�wr; nr� �7:13�Inclusion of the geometric relationships (7.1)±(7.12)yields a determinate system of 13 governing equationsplus 10 geometric relationships in as many unknowns.In these equations we have considered rates of change ofarea fractions with time rather than the area fractions.This approach seems more convenient from an experi-mental point of view, because it requires the experi-mentalist to monitor only the rate at which the varioussubregions expand or contract, without having to mea-sure the actual values of the various area fractions andof the drainage density with changes of the respectiveindependent variables. The form of the functional de-pendence has to be determined through ®eld experi-ments and can vary from site to site according to thelocal topographic and geological circumstances.

8. Conclusions

This work is a sequel to a previous paper by Reggianiet al. [33] in which a systematic approach for the deri-vation of a physically-based theory of watershed ther-modynamic responses is developed. In this approach, awatershed is divided into a number of subwatersheds,called Representative Elementary Watersheds (REWs).Each REW is, in turn subdivided into ®ve subregions:unsaturated zone, saturated zone, concentrated over-land ¯ow zone, saturated overland ¯ow zone, and a

channel reach. A systematic averaging procedure is ap-plied to derive watershed-scale equations of conserva-tion of mass, momentum, energy, and entropy for eachand every subregion of all REW's. The balance equa-tions need to be supplemented with constitutive rela-tionships for the averaged thermodynamic quantities. Inthe present paper, the procedure for the development ofthe constitutive theory is presented and it is illustratedby applying it to a generic watershed.

The paper ®rst introduces a global reference systemand e�ective directions of ¯ow for the ®ve distinct andinteracting subregions contained within the REW. Next,the closure problem is tackled by making use of theColeman±Noll procedure for the exploitation of thesecond law of thermodynamics (i.e., entropy inequality)leading to a set of constitutive relationships which arethermodynamically admissible and physically consis-tent. The procedure is implemented uniformly and in aconsistent manner across all subregions and REWsmaking up the watershed. As a ®rst attempt, linear pa-rameterisations of the mass exchange terms betweenphases, subregions and REWs, are postulated as func-tions of pressure head di�erences, velocities and chemi-cal potentials, in such a way that they do not violate theentropy inequality. These lead, via the entropy in-equality, to equivalent parameterisations of the mo-mentum exchange terms under equilibrium conditions.The non-equilibrium component of the momentum ex-change terms are obtained through ®rst or second orderTaylor series expansions around equilibrium, guided byprevious ®eld evidence. For example, the momentumexchange term relating to subsurface ¯ow uses a ®rstorder expansion, leading to a REW-scale Darcy's law,while the resistance terms relating to overland ¯ow areexpressed in terms of a REW-scale Chezy formula. Forthe channels an equivalent of the Saint-Venant equa-tions has been obtained for a tree-like branchingnetwork of river reaches. The end result is a set of REW-scale balance equations for mass and momentum,comprising 13 balance equations in 23 unknowns.

An additional 10 constitutive relationships are in-troduced, based on geometric considerations, in order toobtain a determinate system of equations; these are alsorequired to satisfy the entropy inequality. The resultingset of 23 equations (13 balance equations plus 10 geo-metric relationships) in 23 unknown REW-scale vari-ables (velocities, depths, area fractions, saturation etc.which do not vary spatially within REWs, and changeonly between REWs), represent a system of couplednon-linear ordinary di�erential equations (ODE). The¯ow within a single REW can in¯uence, and be in¯u-enced by, the ¯ow ®elds in neighbouring REWs,through the exchange of groundwater and soil moistureacross the mantle, and via backwater e�ects along thechannel reaches constituting the river channel network.The above-mentioned coupling amongst the REWs


necessitates a solution algorithm which takes into ac-count the whole ensemble of REWs simultaneously.

The solution of 13 non-linear ODEs for 30 or moreREWs may appear overwhelming. However, comparedto the ®nite element or ®nite di�erence numericalschemes associated with traditional distributed modelswith thousands of nodes, our proposed system ofequations is still very modest. On the other hand,compared to traditional conceptual models of watershedresponse, the parameterisations developed here arephysically-based, and more importantly, they are par-simonious.

The momentum exchange terms derived in this paperare linearised with respect to velocity and/or total headdi�erences. The linearisation parameters A and B arefunctions of the remaining variables of the system.These functions can be either linear or non-linear andneed to be estimated from ®eld experiments or throughdetailed numerical models.

Aside from the summary presented above, the over-riding message to come out of this work is that a com-prehensive set of physically-based, watershed-scalegoverning equations, which respects the presence of astream network, can indeed be constructed based on®rst principles. With appropriate simplifying assump-tions, the equations do give rise to a watershed-scaleDarcy's law and Chezy law; yet the equations are also¯exible enough so that when ®eld evidence warrants itwe can go beyond the constraints of these traditionalmodels and use other more general parameterisationsappropriate to the ®eld situation. Issues such as mac-ropore ¯ow in soils, and rills and gullies in overland¯ow, come to mind in this regard. In this and otherrespects, the derivation of the governing equations alsoprovides the motivation to design new ®eld and remotemeasurement techniques to advance the development ofa new hydrologic theory.

The key to such a development is to keep the theo-retical aspects as general as possible with a minimum ofassumptions. However, being a ®rst attempt, completegenerality was not the goal in this study, for pragmaticreasons. While the overall aim has been to develop ageneral, unifying thermodynamic framework for de-scribing watershed responses, the constitutive theorydevelopment presented has made a number of simpli-fying assumptions to keep the problem at a manageablelevel. In particular, we have focussed on runo� processesat the expense of processes related to evapotranspira-tion. Evapotranspiration makes up over 60% of thewater balance worldwide, and it is important that thetheory presented in this paper be generalised to take itinto account. To fully describe evapotranspiration, weshould include thermal e�ects, movement of both watervapour as well as liquid water in the soil, vegetatione�ects (both root water uptake and plant physiology),and turbulent transport processes in the atmospheric

boundary layer above the land surface. The needed ex-tensions of the theory to include these aspects, which arequite considerable, are left for future research.

9. Future perspectives

In concluding this paper, we feel the need to presentan outline of where we believe the present work may beheading, and what we hope to achieve in the future. Thisseems appropriate, in order to give the readers a far-sighted, although somewhat biased and very muchspeculative vision of the long-term perspectives of ourresearch, and to stimulate their interest and participa-tion in some of the future goals.

In contrast to watershed hydrologists, ¯uid mechan-icians have at their disposal well established equationsgoverning ¯uid movement. Their main research e�orts,at the present time, focus on improving closure schemesfor turbulence, on understanding particular phenomenasuch as mixing, dispersion, density driven ¯ows, naturalconvection, internal waves, and interactions betweenhydrodynamics and chemical and biological processes,and on developing and improving computational algo-rithms for the solution of the governing partial di�er-ential equations. The basic governing equations theydeal with routinely, namely, the point-scale conservationequations for mass, linear and rotational momentum, aswell as energy, were derived roughly two centuries agoand form the very foundation of their work to whichthey could always fall back on for guidance or insight.

This is not the case in watershed hydrology. Thecommon practice in surface hydrology has been to as-semble the equations governing individual hydrologicprocesses (such as in®ltration, overland ¯ow, channel¯ow), which have been derived independently, at smallscales, often in di�erent contexts (e.g., in®ltrationequations derived by soil physicists), and based on moreor less restrictive a priori assumptions (e.g., uniformsoils). Examples include, among others, Darcy's law andits extension to unsaturated ¯ow, the Richards equation,its approximate analytical solutions for in®ltration suchas the Green-Ampt model or Philip's equation, theSaint-Venant equations and the kinematic wave ap-proximation. To describe watershed response, thecommon practice has been to assemble these equationstogether rather mechanically, regardless of the di�eren-ces in context and in scales, i.e., between the scales atwhich they were developed and the watershed scale atwhich predictions are sought. Currently there is noagreed set of conservation equations for mass, momen-tum and energy balances, at the scale of a watershed;certainly not derived within the framework of a singleand systematic procedure, as done in the present work.

In line with our own limited perception of how theresearch ®eld of ¯uid mechanics has developed in the


past and is developing at present, we attempt to give anoverview of the research tasks which will be needed tocomplete the development of a hydrologic theory basedon the set of governing balance equations presentedearlier in this paper. We also try to indicate how thesecan be employed in the future to model general situa-tions related to watershed hydrology, including long-term water balance, hydrologic extremes (¯oods anddroughts), erosion and sediment transport, water qual-ity, and the general problem of prediction of ungaugedbasins (the PUB problem discussed by Gupta andWaymire [20]). Possible steps involved in a generalisat-ion of our approach and its implementation to solvethese problems can be listed as follows.

Inclusion of evapotranspiration. Similar to the Navier±Stokes equations in ¯uid mechanics, we have obtained aset of watershed-scale balance equations for mass andmomentum, supplemented with appropriate constitutiverelationships. These allow us to evaluate the responses ofthe various subregions of the watershed, i.e. in thesubsurface zones, the overland ¯ow zones and thechannel network, to generate space-time ®elds of sto-rages and velocities everywhere across the watershed.Various extensions of the work we have done can beenvisaged. One important example is the inclusion ofboth bare soil evaporation and plant transpiration. Todo this, two things need to be done: ®rst, a new zone hasto be de®ned and added to the ®ve zones of REW de-®ned so far. This zone would represent the atmosphericboundary layer, including vegetation. Averaged equa-tions of conservation of mass, momentum, energy andentropy for the atmospheric boundary layer need to bedeveloped following the procedure outlined and appliedby Reggiani et al. [33]. Exchange of properties betweenthis zone and all the other zones and the external worldwill have to be properly accounted for. Next, followingthe procedure of the present paper, constitutive rela-tionships describing thermal energy exchanges betweenthe various subregions, water vapour di�usion, rootwater uptake, plant physiology, and above all mass,momentum and energy exchanges between the landsurface and the atmospheric boundary layer have to bedeveloped. Our contention is that the governing equa-tions derived above (including extensions to includeevapotranspiration) can form the basis for predictivemodels of watershed response. As a ®rst e�ort in thatdirection, a model of rainfall-runo� response for a bi-furcating stream network has been constructed (Regg-iani et al. [34]), and has been used to estimate space-time®elds of velocity and storage in the network. This modelis currently being extended to include hillslope pro-cesses.

Improvement of constitutive relationships. The con-stitutive relationships presented in this paper are quitebasic, and need to be considerably improved and testedunder various climatic and geographic circumstances,

and the e�ects of variabilities of the hydrologic pro-cesses at scales smaller than the REW need also to beincorporated in the parameterisations. This can beachieved by means of both ®eld experiments and de-tailed numerical models based on traditional small-scalegoverning equations (e.g. Du�y [6]). Indeed, much ofthe previous work which has been carried out on spatialvariability, scale and similarity (Wood et al. [39],Bl�oschl and Sivapalan [2], Kubota and Sivapalan [28],Kalma and Sivapalan [27]) can be placed in this context.The added value of the present work is that with theavailability now of a proper set of balance equations,these need no longer be ad hoc and should be seen asproper closure schemes similar to those investigated in¯uid mechanics. Data collection is important for allmodelling e�orts, to serve as climatic inputs and aslandscape parameters, and to validate model predic-tions. However, in the context of the present work, newtypes of data and new observational strategies areneeded to assist in the development of constitutive re-lationships, especially to quantify the e�ects of sub-gridvariability.

Inclusion of sediment transport, chemistry and biology.The set of balance equations for mass, momentum, andenergy derived by Reggiani et al. [33] govern the trans-port of water within and over the watershed. However,we know that water is also the carrier for sediments,chemical constituents and living organisms through thevarious subregions of the watershed. Therefore, thegoverning equations need to be coupled with generalequations for sediment transport to account for erosion,transport and deposition of sediments. These can laterbe coupled with transport equations for various chemi-cal constituents such as nutrients, in streams, on theland surface, and in the soil pores, for the purpose ofwater quality studies. The same procedure can be re-peated for biological components by combining theequations governing the growth and decline of livingorganisms in the water and in soils (e.g., algae andbacteria) with equations governing stream¯ow, sedi-ments, nutrients (balance of water mass and momentum,and mass transport), as well as temperature (balance ofenergy).

Space-time ®elds, ¯ood scaling, ¯ood forecasting.Numerical models of the governing equations of massand momentum balance at the watershed scale can alsoopen up further exciting possibilities for the investiga-tion of space-time variability of runo� ®elds, includingthe extremes of ¯oods and droughts, and their processcontrols (for reference see e.g. Gupta and Waymire [20],Robinson and Sivapalan [35] and Bl�oschl and Sivapalan[2], [3]). Areas where these will have a direct impact arethe scaling behaviour of ¯oods and the implementation of¯ood forecasting systems. In addition, due to the for-mulation of constitutive relationships which are physi-cally based and meaningful, the need for calibration is


signi®cantly reduced, especially when combined withinnovative ®eld measurements of critical variables.Therefore, this can make a signi®cant contribution tothe classical problem of prediction of ungauged basinsdiscussed by Gupta and Waymire [20].

Long-term water balances. Once the evapotranspira-tion part of the hydrologic cycle is fully incorporated inthe governing equations, and provided they are aver-aged over a suitable averaging time, these can then beused to investigate long-term water balances, and inparticular, the e�ect of climate±soil±vegetation interac-tions on the dynamics of water balance, as an extensionto the pioneering work of Eagleson [7±13,37]. Waterbalance is a simple principle, yet it re¯ects many com-plex interactions which are driven by a competitionbetween the two primary driving forces of gravity andsolar energy, with a mediating role played by soils andplant physiology. All natural hydrologic variability (e.g.,extremes) is underlain by the variability of water bal-ance, and hence understanding water balance variabilityis a fundamental problem. The set of governing equa-tions which incorporate these physical processes can goa long way towards understanding and predicting theobserved natural variability of water balances, and anychanges to these due to human intervention.

Search for hydrologic theory. Finally, we expect thatour approach would deliver appropriate equations andwould then be used to formulate and test fundamentalprinciples which could guide our understanding of theorganisation of landscapes and watersheds (includingclimate, soil and vegetation). Work is needed to in-vestigate possible manifestations of these organisingprinciples on hydrologic variability, and to validatethese against observed data. Examples of such principlesinclude the principle of minimum energy expendituregoverning hydraulic geometry in channel networks (SeeRefs. [36,30]), and the ecological optimality hypothesisgoverning climate±soil±vegetation interactions (SeeRefs. [8±15,24]).

Acknowledgements

P. Reggiani was supported by an Overseas Post-graduate Research Scholarship (OPRS) o�ered by theDepartment of Employment, Education and Training ofAustralia and by a University of Western AustraliaPostgraduate Award (UPA). This research was alsosupported by a fellowship o�ered by Delft University ofTechnology, which permitted P.R. to spend a six monthperiod in The Netherlands. W.G. Gray was supportedby the Gledden Senior Visiting Fellowship of the Uni-versity of Western Australia while on sabbatical leave atthe Centre for Water Research. Centre for Water Re-search Reference No. ED 1177 PR.

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