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Economia & Energia
No 36: Janeiro-Fevereiro 2003  
ISSN 1518-2932

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 Thermodynamics Application to the Evaluation  of Fluvial Networks Equilibrium  - the  river Santo Antonio basin

Project financed by FAPEMIG
Ceres Virgínia Rennó Moreira
Researcher at Fundação Centro Tecnológico de Minas Gerais / CETEC
Omar Campos Ferreira
 Associated Researcher at Fundação Centro Tecnológico de Minas Gerais / CETEC
Paulo Pereira Martins Junior
Researcher at Fundação Centro Tecnológico de Minas Gerais / CETEC

Editor’s Note: Equilibrium of Fluvial Networks.

The article on fluvial networks is an application of the generalized entropy concept proposed by Boltzmann. The entropy function was introduced in the Classical Thermodynamics in order to translate the final static equilibrium fatality in thermodynamic systems, an issue unknown in Classical Mechanics. Therefore, the entropy grows monotonically in the system, reaching its maximum value in the static equilibrium state. In the open systems, as that of the fluvial network that exchanges matter and energy with its surroundings, the static equilibrium is preceded by one or more temporary dynamic equilibrium states (permanent regime), there existing then a sequence of these states that correspond to relative maxima entropy preceding the final absolute maximum value.

The growth of entropy was related by Clausius with the occurrence of irreversible phenomena like the heat exchange under finite temperature difference, diffusion, resistance to movement (friction, viscosity, etc…). In the fluvial network, the irreversibility is due to soil excavation, dispersion of its particles in water, transport of this material downstream, etc…

It is assumed that the total energy is conserved but its availability to cause the evolution of the system decreases while its entropy grows, or that the energy is dissipated, and not consumed, by irreversibility.

This type of application is common in economy planning and even psychology. The macroeconomic model developed by Carlos Feu et al. based on “Economy &Energy” studies incorporates “exogenous” variables based on the entropy notion, like capital depreciation, capital productivity growth and transfers of goods and services abroad

Omar Campos Ferreira in02/08/2003

ABSTRACT 

The theories of average stream fall and least rate of energy expenditure were developed by Chih Ted Yang. Both are specific applications of the entropy law as applied to the analysis of terrain stability / instability. This paper is a report of an analysis of Santo Antônio watershed in Minas Gerais State based in both theories. In this basin the erosion process is a real tendency as it was previously observed with the cartography of the many spots of erosion forms like gullies and land slides, which are abundant. Erosion is a process conducting the watershed to evolve far from a dynamic equilibrium.

RESUMO

             As teorias da queda média dos rios e da mínima dispersão de energia, desenvolvidas por Chih Ted Yang (1971), baseadas no conceito de entropia, foram aplicadas à análise da estabilidade / instabilidade do rio Santo Antônio, pertencente à bacia do rio Grande. Foi confirmada a tendência à degradação da bacia, observada em estudos anteriores de erosão acelerada por voçorocas e escorregamentos e seu afastamento da fase de equilíbrio dinâmico.

 

  INTRODUCTION

            The drainage network analysis using the thermodynamics approach is due to experiments that demonstrate that the dynamic balance concept guides the fluvial systems. We quote the pioneering work by GILBERT concerning the geology of the Utah mountains (USA) in 1877, where the concept of dynamic balance of the landscape evolution was introduced and two conclusions about the fluvial network were presented  - “during the evolution toward its equilibrium condition a natural stream chooses its course of flow in such a manner that the rate of potential energy expenditure per unit mass of water along this course is a minimum” and “ … as the ratio of erosion as a function of the slope becomes equal to the opposition as a function of the type of rock there is an action equilibrium”.

            HORTON (1945) gave a new impulse to the knowledge concerning drainage network by developing a method for its analysis. He developed a classification, ordination and hierarchy system for the fluvial network that comprehends the order laws in the fluvial segments, the number of channels and the number of bifurcations among others. However, the alterations proposed by STRAHLER (1957) for this system are more used presently.

            In 1962, LEOPOLD and LANGBEIN introduced the entropy concept in the fluvial morphology evaluation. According to these authors, by analogy with thermodynamic entropy, it is postulated that a geo-morphologic system (like the fluvial network) is an open system in a permanent state regime. This entails two generalizations about the most probable energy distribution that would be an intermediary state between two states or trends: a state in which the energy dispersion rate is uniformly distributed and a state in which the system does the minimum work.

            YANG (1971),continuing this line of work, has adapted the energy and entropy concepts to verify the type of relationship that exists among the different fluvial orders and to select the ordering that better represented the essence of a drainage network, trying to explain the drainage network formation and the origin of meanders  and the transportation of sediments. He also developed the theory of average rivers fall and the theory of minimum energy expenditure of the fluvial system, affirming that the rivers during their evolution towards an equilibrium condition choose their way so that the potential energy dispersion rate by water mass unit is minimum. However, he warns that the model is applicable to rivers that have attained the condition of dynamic equilibrium, that is, for rivers that during the process of reaching the final static equilibrium have readjusted so that there is a balance between the work done and the carried sediments.

Then the Yang (1971) methodology was applied to the river Santo Antonio that is a tributary of the river das Mortes and both are part of the river Grande basin. The objective was to verify the adjustment degree of its present longitudinal profile to the calculated and equilibrium longitudinal profiles as well as to verify if this methodology could attest the trend to erosion already verified in previous studies.

           Stability analysis that indicate trends and drainage changes are an  important tool to back actions that, for example, aim at restoration of a river, that is, re-establishing the structure, function and dynamics of its ecosystem {ISRWG (1998)}.

 

METHODOLOGY

 The work was carried out on 1:30 000 aerial photos of 0- 385 flight and with 1:50000 and 1:100000 topographic sheets from FIBGE of Jacarandira, Resende Costa, Tiradentes, and  São João del Rei.

           From the detailed mapping of all visible channels of the drainage network in the aerial photos and their transposition to topographic bases, data regarding hierarchy of the drainage network, altitude, length and drainage area of all fluvial orders were obtained for the purpose of calculating the Horton – Strahler equations and construction of their diagrams and calculating the Yang equations and construction of the present, calculated and equilibrium longitudinal profiles of the river Santo Antônio.

 

Analogy equations for a hydrographic basin to be considered an open thermodynamic system in a quasi-stationary regime.

 

            The methodological base is founded on the PRIGOGINE (1967) theorem for a classic thermodynamic system tending to a permanent regime that assumes:

           The entropy inherent to the system tends to a maximum value compatible with the restrictions imposed by the surroundings.

            The analogy equations, according to Yang (1971), define:

Hu –the average loss of potential energy by mass unit of water for all water courses of order u.

Yu – the fall (level difference) between the source and mouth of the water course of order u .

Zm –    the total fall between the water course source of order u = 1 and the mouth of the water course with the largest order in the basin (in the case analysed, m=7)

Hu= g Yu                                       (1)

           where g is conversion factor between energy and fall.

           By thermo-mechanics analogy, the absolute temperature in a thermal system is equivalent to a raising in a fluvial system and the thermal energy in a thermal system is equivalent to the potential energy in a fluvial system. We have then:

            The entropy variation of the system at temperature T due to heat exchange dE:

DS = ò dE / T

            In the mechanical case (by analogy), for the fluvial network:

DS = ò dH / Zm = g ò ( dYu / Zm )

            For the water course of order u:

DS u = g òu dYu / Yu

The probability that a determined energy loss occur in the course of order u is: 

p = Hu / Hm = Yu / Zm                                        (2)

The average entropy variation in the courses of order u is:

DSu = g òu dYu / Yu = g òu Zm dp / Zm p = g ln pu + constant                          (3)

             The most probable energy distribution in the basin in a stationary regime is the one that maximizes the function.

DS = åu DSu = g åu ln pu + Constant  provided åu pu = 1                            (4)

            It is a classical problem of determining a conditioned extreme that is solved by Lagrange’s undetermined multipliers method. It can be demonstrated that the maximum occurs for p1 = p2 = p3 ......= pm

            Observing equation (2), it results the law of average fall equality of each course order that can be expressed by:

Yu =Yu+1 = constant                    (5)

 

Horton – Strahler’s empirical laws

 

             Adopting Strahler’s water courses ordering, and Horton’s law concerning the number of channels, the average length and the average steepness of channels are given by equations :

                  ln Nu = A – Bu

                  ln Lu = C – Du

                  ln Su = E – Fu

                  ln Adu = M – Nu

where:

     u     =   order number of channel

     Nu   =   number of channels of order u

     Lu    =   average length of channels of order u

     Su    =   average steepness ( Yu / Lu) of channels of order u

     Adu =   average area of the drainage channels of order u

From these laws the following expressions are:

eB = Nu / Nu+1 (bifurcation ratio)

eD = Lu / Lu+1 (average length ratio)

eF = Su / Su+1 = ( Yu / Lu) / ( Yu+1 / Lu+1)

(concavity of the course profile of order u)

eN = Adu / Adu+ (drainage area ratio)

           The approaching to the permanent regime ( maturity of the basin) can be evaluated  by the equality of falls (equation 5). The other form of evaluating the maturity is to compare the basin’s calculated longitudinal profile with the equilibrium profile. The profile is described by the equation:

åu  Yu = f (åu  Lu)

            From the empirical laws the total fall and the total course until order u are calculated , namely:

åu Yu = e(C+E) åu e – (D+F)u

           åu Lu = eC åu e –uD                        (6)

           When the drainage basin reaches,

               

e – (D+F) = 1  e  åu Yu =u e (C+E)   (7)

           The calculated profile is described by equations (6) and the equilibrium profile is described by equations (7) with the same values of åu Lu used for the calculated profile. The construction of a graphic from the mentioned equations gives the longitudinal profiles of the drainage network. The maturity is evaluated by the difference between the calculated and equilibrium profiles.

  SANTO ANTÔNIO RIVER BASIN

           The river Santo Antônio basin comprises an area of 500 km2, situated at the Vertentes mountain whose main crest line divides the waters of the São Francisco and Grande rivers.

            The morphology of the  area is characterized by hills elaborated on the granite base – gneissic in the central region of the basin – and by more elevated hills in the amphibolitic areas around them. The drainage network is dendritic with parts of the main river and of the tributaries controlled by NW and NE predominant structural directions.

 

Granitic and amphibolitic outcrops condition waterfalls and rapids found on the top course of the river Santo Antônio.

 

           Studies by CETEC (1989) have verified in this basin a large concentration of erosion focuses – a total of 754 focuses – of which 447 have been characterized as gullies (Figure 1). MOREIRA (1992) has shown that the basin has been submitted throughout all its geologic history to several accelerated erosion episodes that have been recorded by the presence of gullies and land slidesand by correlative deposits in several slope and drainage network points.

             These characteristics have been decisive for choosing the basin because they permit the evaluation of the erosive process impacts on the drainage network and the capacity or lack of capacity to adjust itself to these processes. In Table 1 the basic data of the river Santo Antônio hydro-graphic basin are listed.

 

Figure 1:

 

 

Table 1 – Morphometric data of the river Santo Antônio drainage network

 

Order

1

2

3

4

5

6

7

Variabless

 

 

 

 

 

 

 

Nu - N. of channels

3 050

803

189

42

9

2

1

Yu – Average  falls(m.)

39,29

27,92

25,19

34,57

41,33

56,00

5,00

Total length of channels (m.)

791 160

343 800

172 290

101 700

61 710

47 280

2 820

Lu- Average length of channels(m.)

259,40

428,10

911,60

2 421,40

6 856,70

23 640,00

2820,00

Area of basins (ha)

23 320

35 723

43 130

47 580

49 640

51 663

51 805

Adu –  Average  area of basins (ha)

7,64

44,48

228,20

1 132,85

5 515,55

25 831,50

51 805,00

Su – Average falls (0/00)

50

65

27

14

6

2

1

 

The  Horton - Strahler Diagram (Figure 2) was constructed using these data .

 

Figure 2Horton – Strahler Diagram of the Santo Antônio river basin, MG

            Then the Yang equations are calculated and  Tables 2, 3 and  4 are obtained for calculating the calculated and equilibrium profiles

.

            Calculation of the calculated profile (equations 6 e 7):

Calculated vertical fall:

Zm = e(C+E)  Sum e - (D+F) u

Calculated horizontal distance:

Xm = eC Sum e-uD

Calculation of the  equilibrium profile:

m = e(C+E)                                   (8)

Constants:

          C+E = -2,26 + 7,37 = 5,11

          e(C+E)  = 166

          eC = 0,104

          D+F = -0,614 + 0,769 = 0,155

          e-(D+F)  = 0,856    

Equilibrium vertical fall:

Z'= m e (C+E)

Equilibrium vertical distance:

X u  =  Xu

Table 2 - Calculated Vertical fall

Order

e-(D+F) u

Su7   e -(D+F) u

e (C+E) Se-(D+F) u

Zu - Z  u+1

S (Zu -Z u+1)

1

0,856

3,949

654

142

142

2

0,733

3,093

512

121

263

3

0,628

2,360

391

104

367

4

0,538

1,732

287

89

456

5

0,461

1,194

198

77

533

6

0,395

0,733

121

65

598

7

0,338

0,338

56

56

654

Table 3 – Calculated horizontal distance

Order

e- uD

Su7 e -uD

eC S e -uD

Xu - X u+1

S (Xu - X u+1)

1

1,85

158,1

16,50

0,19

0,19

2

3,41

156,3

16,31

0,37

0,56

3

6,31

152,8

15,94

0,65

1,21

4

11,7

146,5

15,29

1,22

2,43

5

21,5

134,8

14,07

2,25

4,68

6

39,8

113,3

11,82

4,15

8,83

7

73,6

73,5

7,67

7,67

16,5

Table 4 Calculated profile (eqs. 6 and 7), equilibrium profile (eqs. 7 and 8), and observed profile (river Santo Antonio profile– data obtained from the IBGE maps).

Calculated profile

Equilibrium profile

Observed profile

Zm (eq. 6)

Xm (eq. 7)

Zm (eq. 8)

Xm (eq. 7)

Zm (map)

Xm (map)

142

0,19

166

0,19

129

0,16

263

0,56

331

0,56

221

0,42

367

1,21

497

1,21

303

0,99

456

2,43

663

2,43

416

2,50

533

4,68

828

4,68

551

6,76

598

8,83

994

8,83

735

21,50

654

16,50

1.160

16,50

752

23,20

Note: Z is expressed in foot (0,305 m)  and X in miles (1.609 m)

 Figure 3, constructed with data from tables 2,3 and 4 permits to compare the observed longitudinal profile with the calculated and equilibrium ones.  The longitudinal profiles, both observed and calculated, have a good aggregation, and we point out the advantages of the latter due to the application of equations 6 and 7. According to Yang, the advantages are due to the fact that the equations

 “são derivadas de parâmetros que representam as características de todo o sistema de drenagem, em contraposição ao perfil observado, que é construído somente com os dados do canal principal”.

However, the equilibrium profile is very far from the calculated profile, indicating that the analysed fluvial system has not reached the dynamic equilibrium condition, since the fall ratio of the river Santo Antônio is larger than one. The equilibrium profile, having a curvature below the calculated profile, corroborates the trend already observed in the basin, namely the trend to accelerated degradation or erosion. Also due to the law of river falls and to the distribution of the potential energy dispersion along the course of the rivers, it is observed that the concavity of the basin is a determining factor regarding the formation of channels network. This characteristics can therefore explain the trend in the studied basin to the high occurrence of gullies. In the analysed case it has occurred an increase in the concavity of the longitudinal profile that, according to Yang’s laws causes an increase in the frequency ratio (or an increase of the number of channels of inferior order) and a decrease of the connecting ratio (or a decrease in the length of the channels of inferior order). The gullies can start at the slopes, connecting themselves to any drainage order or at the bed of the order 1 channels. In the first case, the gullie´s main channel can be considered as a new order 1 channel and in the second case the installation of a new gullies only causes the expansion of one order 1 channel. In the river Santo Antônio basin, 70% of the gullies contribute to increasing the number of order 1 channels. They also contribute to reducing the length of the channels of this same order since after their start and connection to an order 1channel, the stretch downstream the connection changes from order 1 to order 2.

 

Figure 3 – Longitudinal profiles of the river Santo Antônio 

  CONCLUSION

            Application of Yang’s methodology to the river Santo Antônio  basin has confirmed the trend to disruption of this basin observed in previous studies and has shown that the basin has not yet reached the dynamic equilibrium phase. The causes of this fact can be associated with episodes of regional raising due to tectonic reactivations that were raised by Moreira(1992).

           Yang’s methodology is easily applied to any hydrographic basin with the available geo-processing techniques and can be used in the evaluation of impacts in dams for the selection of basins that shall have priority in studies concerning environmental conservation or to subsidized actions that, for example, aim at the restoration of a river.

 

BIBLIOGRAPHY

CETEC.1989. Caracterização Ambiental da bacia do rio das Mortes, MG. Belo Horizonte, relatório inédito.

FISRWG - Federal Interagency Stream Restoration Working Group. (1998). Stream Corridor Restoration: principles, processes  and practices. Usda.gov / stream-restoration / copies. htm

GILBERT, G. K. 1877. Report on the Geology of the Henry Mountains. US Department of the Interior.

HORTON, R. E. 1945. Erosional   development of streams and their        drainage basins: hydrophysical approach to quantitative morphology. Bulletin of the Geological Society of America, v.56, p. 275-370.

LANGBEIN, W. B. ; LEOPOLD, L. B. 1964. Quasi - Equilibrium States in Channel Morphology. American Journal of Science, v. 262, p. 782-794.

LEOPOLD, L. B.; LANGBEIN, W. B. 1962.  The concept of entropy on landscape evolution. US Geological Survey Prof. Pap., 500-A.

MOREIRA, C. V. R. 1992. Fatores condicionantes das voçorocas na sub-bacia do rio Santo Antônio, Bacia do rio Grande, MG. Belo Horizonte, IGC-UFMG, dissertação de mestrado, 163 p.

PRIGOGINE, I. Introduction to Thermodynamics of Irreversible Processes. 1967. John Wiley, New York, 147p.

STRAHLER, A. N. 1957. Quantitative  analyses of watershed geomorphology.   Trans. Amer. Geophys. Union, v. 38(6), p. 913-920.

YANG, C. T. 1971. Potential Energy and Stream Morphology. Water Resources Research, v. 7(2), p. 311-322.

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Application of thermodynamics to the evaluation of equilibrium of fluvial networks – the river Santo Antônio basin. The analysis instrument, based on thermodynamics, has multiple applications in other areas, including economy. In the present case, the average river falls and the minimum energy dispersion theories developed by Chih Ted Yang (1971), based on the entropy concept, were applied to the stability/instability analysis of the river Santo Antônio, part of the river Grande basin. It was confirmed the trend to basin erosion, observed in previous studies relative to accelerated erosion by gullies and land slides and its distance from the dynamic equilibrium phase.