METHOD OF CALCULATING THE MAXIMUM WATER FLOW IN THE MOUNTAINS

193

METHOD OF CALCULATING THE MAXIMUM WATER FLOW IN

THE MOUNTAINS

YAVKIN V.1_, GOPCHENKO E.2

ABSTRACT. - Method of calculating the maximum water flow in the mountains. The Befani-Gopchenko model is suggested to calculate the maximum water expense in the absence of hydrological observations. The model is a combi-nation of reduction, dimensional and critical intensity methods used in normative practice in the majority of world countries. Statistical analysis of time-series obser-vations is applied, and the methods to define parameters and coefficients in the model are suggested. Their spatial distribution corresponds to general geographical properties of hydrographic characteristics in the mountains and is offered to calculate the maximum flood runoff. The applied method contains useful information to help prepare regional normative acts to define major constructive components of hydrological estimations of maximum water runoff.

Keywords: engineering calculations of maximum runoff, flood layer, module of maximum runoff, relief.

1. INTRODUCTION

A system of estimation, forecasting or calculation of flood hydrographs includes methods that base on cognition of physical and statistical essence of preci-pitation’s temporal and spatial distribution, and water volumes transformation on the surface and in soil container of slope and channel drainage basin.

Physic-mathematical modeling of rainfall flood runoff is rather widely re-presented by complex models with lumped parameters that describe transformation of flood waters on the slope with the linear function.

The use of standards in estimation of maximum runoff faces essential dis-crepancies between the values of predicants calculated in accordance with pre-scribed methods for high waters or floods, and the materials of field observations. It is especially evident in cases where the parameters of underlying surface are es-sentially mosaic or simply different from the so-called “typified” for the specific region. Analysis and generalization of the whole history of development of flood maximum runoff estimation allow for maintaining that there is an urgent need in development and introduction of new normative directions based on two-operator model that is distinctive for sufficient universalism, preservation of genetic stages

1

Chernivtsi Yuriy Fedkovych National University, Faculty of Geography, 58012 Chernivtsi, Ukraine E-mail: [email protected]

2

194

of maximum formation in chain sequence, and, what is more, operation with nece-ssary, tried and tested nomogram and cartographic materials.

A new version of computing method to define maximum runoff is suggest-ed. This was substantiated basing on the concepts by E.D. Gopchenko. Up-to-date initial materials for maximum runoff are used, while the suggested estimation basis stays the same irrespectively of the watershed size.

Estimated values of flood flow maximum expense and layers are defined by flow duration smooth curves extrapolated by binomial curve.

The values of water maximum expense and corresponding flow modules of different duration vary within wide range depending upon physic-geographic cond-itions, water intake area, degree of regulation, forest coverage, basin bogging, etc.

2. DISCUSSION

A new version of calculation procedure to determine the maximum flow is suggested. The up-to-date initial materials for maximum flow are used while the estimate basis stays common regardless of the size of the catchment (Review of Risk-Based Prioritization, 2004;Rozhdestvensky, 1997,).

All formula used herein to calculate the maximum flow are divided into two groups. The first group includes those based on single-mode hydrographs of floods described by the equation:

_{m n} p

m

m k k

T t

Y k q

0 0

1

 , (1)

where

q

_m is the maximum flood runoff module;

Y

_m stands for layer of flood run-off; and

k

₀ is the coefficient of slope transformation of floods.

0 0

1 1

T n n

k   ; (2)

where

n n1

stands for coefficient of temporal irregularity of slope runoff;

T

₀ is the duration of the slope inflow; and

k

_m represents flood hydrograph form’s

transformation coefficient.

n n m m

k_m 1/ 1; (3) where

m m1

–coefficient of channel runoff’s temporal irregularity during floods;

195

n p n

T t T

k  0 ; (4) where tp – time of flood waves travel to cross sections;

T

n – floods duration.

The second group includes formulas based on channel isochrones model: _{ã F}

p m

m

k

t

Y

q







, (5) where



- participation coefficient of the

Y

_m in the formation of the maximum water expenses





  

0

0 0 T

t t

t

dt q

dt q

p



, (6)

where

q

_t



– slope inflow reduction hydrographs’ ordinates;

k

_ã – hydrographic coefficient, where

- with t_pT₀

    

  

 







p p

t

t cp t

t t

ã

dt q B dt B q k

0

0 _{, (7)}

- with tpT₀

    

  

 







0 0

0 0

T

t cp T

t t

ã

dt q B dt B q

k (8)

where

B

_t – watershed width as by isochrons channel travel; B_cp – watershed average width as by isochrons channel travel; and



_F – coefficient of channel/floodplain floods regulation.

Depending upon the proportion between the time of channel travel and slope inflow: - with t_pT₀







p

t

t t t ä

m V qB dt

Q 0

196 - with tpT0





 0 0 T t t t ä

m V qB dt

Q



, (10) where

V

_ä – speed of channel travel;



_t - channel/floodplain floods regulation function.

In a settlement

F p m m

_T

t

q

q





















0

, (11) where

q

_m



- slope inflow maximum module

_m Y_m kY_m T n n q 0 0 1 1   

 , (12)

      0 T tp



– transformational function of flood subsidence under the influence of channel travel:

- with 0

0  T t_p       0 T t_p



= 1,0; (13)

- with 1,0 0  T tp







n p p T t n m n m T t                   0 1 1

0 1 1

1 1



; (14)

- with 1,0 0  T t_p





_                           1 0 1 1 1 1 0 0 1 1 1 1 m p p p t T n m m n m m t T n n T t



; (15)

- with t_p T₀

      0 T t_p



= 0. (16)

F



- coefficient of channel/floodplain floods regulation.

me-197

asured within the system adopted by the Hydrometeorological Service. However, methods of numerical solution of the initial equations of non-measurable parame-ters were suggested in the last years. To improve the quality of calculation schemes it would always be desirable to reduce the number of basic parameters. Proceeding from (12), we can put in the structure of the bulk formula as follows:

_{m n} p

m

m k k

T t q q 0 1 

 (17)

from which the coefficient of total reduction m m

q q

 will be: ( ) 1 F f T t k k q q n p n m m m   

 , (18)

where F is the river basin area (km²).

As to (15), it can also be resuced to (18) if numerator and denominator of (15) is multipled by

k

₀, in accordance with (11). The (15) will therefore take the form of _{ã F} p m m

k

t

T

n

n

q

q

0





1







, (19)

with (18):

(

)

1

0

_k

_f

_F

t

T

n

n

q

q

F ã p m m













. (20)

As to (11):

  m m q q ) ( 0 F f T t F p _ 

   



_



. (21)

All three levels are likely to be quite complex in terms of their realization. To simplify the calculation scheme, we can limit ourselves, for example, to single-modal reduction hydrograph as revised:

                m n m t T t q

q 1 . (22) Upon integration of the expression we will have:

_{m m} T_n m m q Y 1 

 . (23) From (23) the runoff maximum module

q

_m is equal to:

n m m T Y m m

198

n m

m

T T

n n m m

q

q 0

1 1

  

 . (25)

With notation for runoff hydrographs form coefficient being

n n m m

k_m

1 1

 

 , and

channel/floodplain regulation parameter n n

T T

k  0 _{, we obtain:}

k k f(F)

q q

n m m

m  

 ; (26)

Compared with previous structures, inclusive of (17), expression (26) has fewer calculation values, majorly those of no high accuracy.

Taking into account (12), the (26) can be revised as follows:

q

_m



k

₀

Y

_m

k

_m

k

_n. (27) It is (27) that will be suggested to be applied to normalize characteristics of rain floods within the rivers of the Karpato-Podilskyy Region. Such parameters as

q

_m,

Y

_m and

k

_m can be established proceeding from the materials of observati-ons, though two parameters -

k

₀ and

k

_n - will still remain unknown. The solution of (27) will be very simple if method of successive approximations is used (with imposition of limitations on the channel/floodplain regulation

k

_n).

The

k

_n is in the first turn taken as 1, thus meeting the condition of the absence of effects of channel/floodplain regulation. Thus, on the first stage:

m m

m

Y k

q

k0  . (28) Since the coefficient of channel/floodplain regulation is

k

_n



1

.

0

, all the values will the more be raised to power, the greater the catchment area is (Freeze, 1980; Hennrich K.,1997; Takagi F., 1980). To exclude a component conditioned by the adoption, regardless of the size of catchment areas,

k

_n



1

.

0

it is advisable to build a dependence

k

₀



f

 

F

. The dependence is described by the equation:

k₀ k₀elg(F1); (29) where

k

₀



- coefficient of slope transformation at

F



0

.

199 lg( 1)

0

0   



 F

n e

k k

k  . (30)

With availability of (30) we can calculate

k

₀for all watersheds (in second approximation)

n m m

m

k k Y

q

k₀ . (31)

Further obtaining of

k

₀ is subject to spatial generalization. The same pattern is valid to realize equations (30) and (31), putting them as follows:

q

_m



q



_m

f

(

F

)



k

₀

Y

_m

f

(

F

)

. (32) On the first stage, f(F) is taken as 1, and

k

₀ shall be equal to:

m m

Y q

k₀  . (33) Next, the empiric relationship

k

₀



f

(

F

)

is constructed with which we would determine

k

₀, that is, represent it by (29). Later, f(F) will be determined

f

(

F

)



e

lg(F1); (34) and the precision of

k

₀is carried out.

3. CONCLUSIONS

Estimated values of flood flow maximum expense and layers are defined by flow duration smooth curves extrapolated by binomial curve.

The values of water maximum expense and corresponding flow modules of different duration vary within wide range depending upon physic-geographic conditions, water intake area, degree of regulation, forest coverage, basin bogging, etc.

By the character of 1%-duration runoff layer distribution in the Karpato-Podillia Region, the following areas are outlined therein: (Gopchenko Y., Yavkin V, 2014).

I – Peredkarpattia 100 - 200 mm II – North-Eastern

Slopes of the Carpathians 240 - 250 mm III – South-Western

200

The biggest 1%-duration runoff layers (over 300 mm) are observed in the upper streams of the Bila Tysa, Prut and Solotvynska Bystrtsia, while the highest – in the upper stream of the Stryi River basin (Yavkin, 2005).

REFERENCES

1. Gopchenko Y., Yavkin V. Rows of Observation of Peak Discharge and Drainage Layers of the Rivers of the Karpato-Podillia // Topography and Climate: Proceedings of International Scientific Symposium. (23-25 Oct. 2014).- Chernivtsi: Tehnodruk, 2014.- 100 p.- P. 56-58.

2. Freeze R. A Stochastic-Conceptual Analysis of Rainfall Processes on a Hillslope.- Water Resourc. Res.- 1980, 16, No.2.- P.398-408

3. Hennrich K., Schmidt J. Dikay R. Regionalization of Geomorphometric Parameters in hydrological Parameters in Hydrological Modeling Using GIS // IAHS Pudl. - 1999.No 254.

4. Review of Risk Based Prioritization / Decision Making Methodologies for Dams. US Army Cops of Enginners. 2004, P. 42.

5. Rozhdestvensky A.V. Present State and Prospects of Hydrological Computations Development. Runoff Computation for Water Projects. / In: Proceedings of the St. Petersburg Symposium. - Paris, UNESCO, IHP-IV Project M-14 / Technical Documents in Hydrology, No. 9, 1997, pp. 9-15.

6. Takagi F. Matsubayashi U. On the Averaging Process of Runoff Characteristics within Watersheds // Proc. 3d Int.Symp. Stochastic Hygraul. / Tokyo, 1980, - P. 263-274.