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WATER BALANCE THEORY IS MORE THAN 340 YEARS OLD
Iofin Z.
Doctor of technical sciences, Ph.D., Professor, Institute of Mathematics and Rational usage of natural resources,
Vologda State University
Abstract
Since Pierre Perrault's publications (1674) and till nowadays water balance equation was used despite it resulted to serious errors in calculation, due to one of its terms absence. This article provides linear-correlation model, which does not have this drawback.
Keywords: linear correlation model, infiltration capacity, depth of absorption into the soil, water balance equation, soil water storage, soil moisture, capillary fringe.
Introduction
Water balance is the basis for assessment of water resources in different river basins and territories. It is one of the parameters for hydroeconomic calculations. The results of such calculations determine the capacity of reservoirs, the required water inflow from rivers and reservoirs, irrigation regime evaluation, assessment of water resources of the territories and so on. The attempt is made in this article to acquaint the professional community of hydrologists and specialists in related fields of knowledge with a new approach to water balance theory and water balance equation.
The suggested approach differs from the existing one in the running value of the regression coefficient of the correlation between the runoff depth and precipitation depth. The variable regression coefficient is included in all the calculation formulae within the linearcorrelation model introduced by the author. In fact it regulates all the correlations in the model.
The existing water balance equation by Pierre Perrault (1674) can calculate only one water balance parameter. The suggested approach based on the dependence between the river runoff depth and precipitation depth is a guarantee of calculating more than ten water balance parameters (Iofin 2013). They are: Rt- depth of absorption into the soil before water formation, Ro -depth of surface detention, Rz - depth of absorption into the soil during flood recession, E= Rt+ Ro+ Rt - depth of potential natural water loss, Eb - depth of free-water-surface evaporation, Ep - transpiration depth for the river basin as a whole, U - infiltration capacity into the ground water, Vd - soil moisture, h - soil water storage, Kv - layer of capillary fringe,2b^ - depth of the subsurface runoff, Uy - layer of surface water exchange and Ud - layer of underground water exchange between neighboring river basins.
Such a difference in the number of calculated water balance parameters is not accidental. This is primarily due to the constant regression coefficient in the equation by Pierre Perrault (1674). In this equation the regression coefficient of the dependence between the
river runoff depth and precipitation depth is equal to unity. This is the equation by Pierre Perrault:
P = Y + E (1),
or
Y = P - E (2)
In equation (1) the regression coefficient for Y is equal to unity, k =1.
The situation is not so clear in case of the constant coefficient of regression of the straight line correlation (2). It happens that the same amount of precipitation falls in the catchment areas, but they may be different in size. As the coefficient of regression is equal to unity, river runoff depths in these catchment areas turn out to be equal too. However, they cannot be the same. On any surface there are unconditional precipitation losses which result in the real regression coefficient and in different river runoff depths as well. Also, equation (1) does not include the parameter of extensive underground basin. Therefore, equation (1) cannot be called the equation of water balance. It does not show precipitation loss by all the components of water balance. The serious error in equation (1) may occur due to the lack of precipitation loss by groundwater recharge.
Many scientists (Voeikov 1884; Oldekop 1911; Velikanov 1940; Alekseev 1962; Befany 1989) were aware of the fact that the structure of equation (1) must be changed. A large group of hydrologists and hydroeconomic specialists (Budyko 1956; Andreyanov 1960; Mezentsev 1969; Bulavko 1971; Babkin 1982; Koronkevich 1990) studied the problems of water balance equation. They suggested various types of water balance equations, but all the scientists returned to equation (1) regardless of their differences. These attempts were unsuccessful due to the fact that there was no common way to calculate infiltration in different catchment areas in the absence of hydrogeological regime observations. Hence, it became impossible to prove the comprehensive structure of the water balance equation and improve the accuracy of calculation with the help of equation (1).
Materials and methods
The presented results were obtained after the analysis of the observation data from 53 catchment areas of Russian water balance stations, 46 catchment areas in Northwest Russia, 22 catchment areas in Eastern Cuba and 6 catchment areas in Italy.
It is hard to believe that since the time of Pierre Perrault, little has been changed in the water balance theory and the structure of the water balance equation. Still, the equations introduced by Velikanov (1940) and Befany (1989) suggested more unbiased structures of water balance equation:
P = E + Y + U + d (3)
Y = P - E - Un+1 (4)
where P = depth of precipitation in the catchment area, mm;
Y = river runoff depth, mm; Un+1 = infiltration capacity into groundwater which is lower than the river bed, mm;
E = depth of potential natural water loss, mm; d = layer of water exchange between the adjacent basins under the ground, mm.
In equations (3) and (4) Un+1 and d are unknown quantities. They certainly can be calculated using hy-drogeological monitoring observation data. But, these monitoring observations do not cover all the river basins and the majority of ordinary river basins do not contribute such observations. Therefore, in case there were no monitoring observation data concerning groundwater levels in the wells it was impossible to calculate infiltration in the whole catchment area. Even if such observations were present, then another even more difficult problem of representing the spatial variability of this value would arise. For these reasons, the revision of the structure of equation (1) also remained impossible.
A new linear-correlation model of water balance based on the straight line equation "runoff-precipitation" (Iofin 2013) was introduced while searching for a way out of this difficult situation. The method of least rectangles (or the method of equally provided values) is used to estimate the quality of such a correlation. This method leads the results to functional correlations. The least rectangles method substantiated by Alekseev (1962) is easy to use. One should arrange the figures corresponding to the predictands and the predictors in decreasing order.
The "runoff-precipitation" graphs are made for two most common types of the flat land run off: perched and banked-up slope runoffs. In one of the works (Befany 1989) it is proved that the perched runoff occurs due to the surface water formation, i.e. due to heavy precipitation which exceeds surface-water infiltration. The banked-up slope runoff appears in the areas with shallow groundwater when the latter rises to the surface. As these two types of the slope runoff are formed under different conditions, their water balance parameters are calculated with the help of different mathematical expressions. Let us consider the mathematical expressions which are characteristic of these types of the runoff.
Some explanations and final expressions are presented here, the development of the correlations are omitted.
In the article the following straight line equations are regarded as the variants of water balance equation:
Y = P - b4 , mm (5)
Y = k4P - b4 , mm (6)
Equation (5) represents the existing water balance
equation with the regression coefficient equal to unity. Equation (6) is a suggested variant of water balance equation (Iofin 2013).
d
Figure 1. Graphic presentation of water balance parameters (with negative b)
In recent works (Iofin 2013) there have been received the mathematical expressions to calculate water balance elements on the basis of this equation. There is also a graphic presentation of water balance (Fig. 1) for the perched slope runoff.
According to Figure 1, the runoff starts with some retardance in relation to the beginning of atmospheric precipitation. This retardance is represented by segment b4; according to the genetic runoff theory it is represented by the sum of two parameters: Rt and Ro, that is
b4 = Rt + Ro , mm (7)
where Rt = accumulation of the depth of absorption within the period of the runoff existence, mm;
Ro= absorption after the runoff period based on the amount of water which remains in closed depressions, mm.
Apart from these types of losses under the conditions of the perched slope runoff when the absolute term of equation (6) b4 is negative it is possible to calculate Rt and Ro with the help of the dependency graph in which b4 relates with the geomorphic factor 0. The physical process which is represented as segment oB includes one more type of seepage loss - R It is a type
of seepage loss that takes place within the period of runoff recession after the process of water formation. According to Fig. 1 and the existing equation, R is not in-
cluded in the water balance equation. Not taking into
account this parameter and the value of infiltration capacity results in serious inaccuracy in calculating the potential natural water loss and water balance:
RT = h
i-fc4
(8)
In this mathematical expression both b and k are the parameters of water balance equation (6). Thus, the sum of the depths of absorbtion into soil represents the depth of potential natural water loss:
i-fc4
E = RT + R0 + RT = b4 + b4^ = ^ (9) The mathematical expression for the infiltration
capacity is as follows:
Un+1 =
Here, Y4 = runoff depth
i-fc4
i-fc4
(10)
: relative part of loss in the runoff.
The act of multiplying by b4 and Y4 concerning the part of loss in the runoff represents the absolute value of the corresponding loss. Mathematical expression (7) shows absolute precipitation loss by absorption before the process of water formation and after the runoff period. Mathematical expression (10) shows the absolute loss of the runoff depth by infiltration.
Figure 2. Graphic presentation of loss ofprecipitation and river runoff under the conditions of banked-up slope
runoff (with positive b)
к
4
44
к
4
к
4
Let us consider the second type of slope runoffs -the banked-up slope runoff. In contrast with the preceding variant with b4, the graph is changed when b5 is positive. The explanation of the changes is in the fact that the ground water table is very close to the surface. That
is why the soil capacity has already become maximum by the runoff beginning.
It is necessary to introduce such concepts as active and passive water formation. Here, passive water formation means appearance of water on the ground sur-
face without forming any runoff. Active water formation represents the runoff which was accumulated on the ground surface.
The absorption loss within the recession period
A,
5 , where b5 is positive, can occur during the processes of both passive and active water formation.
» i - k
Segment 5 is formed during the process of
passive water formation when the depressions of microrelief are filled at the moment of minimum absorption in its decrease. This decrease of absorption is formed by microrelief, soil and rainfall intensity (Be-fany 1989). Microrelief is filled with the excess water on the surface. The excess water is connected with the rate of the absorption decrease and microrelief.
Active water formation (segment BF, Fig. 2) starts when the microrelief depressions which prevent from the runoff are filled.
Active water formation presents water storage which forms the runoff. Rectangle BFGH (Fig. 2/ is re-
b, 1-k
stricted by value 5 and occurs only due to the
process of passive water formation. At the same time water formation process can occur when the ground water table is very close to the daylight surface. In this case the soil water capacity is maximum by the beginning of water formation and it forms the loss repre-
b ^
5
sented with segments b5 and 5 .
Triangle BEF is the zone of passive water formation. Triangle BED is the zone of active water formation. The subsurface runoff appears in this type of banked-up slope runoff. It is formed when friable soil is flooded with ground water. It occurs in one part of the catchment area or in the whole catchment area when the water table which is very close to the surface can reach the daylight surface in case of rainfall intensity.
,1 + h
b
A new mathematical expression
ap-
pears with this type of the runoff. It has not been used in the linear-correlation model before. Y-axis of the line of the subsurface banked-up slope runoff is zero at the point of the runoff beginning and 2b5 at point P. The point is measured from the line of the conditional separation of the processes happened on the surface of the
catchment area Y5 and &s(1 fcs)and the processes of the
subsurface runoff.
The depth of potential natural water loss in such a banked-up slope runoff does not change its mathematical expression and is presented as:
E = b5 + b5
1 - k
5
h k5
Infiltration capacity is calculated according to the following dependence:
P - (k5P + b5) - b = U
t
b
P( 1 - k5) - b5 —5 = U
P (l - k5 )-b5l+k5
, mm (12)
, mm (13)
= U
P - Y -
b5 + b5
1 - k
= U
, mm (14)
(15)
V 5 J
where U = infiltration, mm.
At first sight it may seem rather problematic to use only linear correlations in the model. The linearity both in the model itself and in the equation of dependence between the river runoff depth and precipitation depth can prevent scientists and experts from adoption the "runoff-precipitation" graph which is always straight. It is impossible to deny that the practice of hydrology has demonstrated curvilinear relation graphs along with rectilinear.
However, it is proved by the author of the article (Iofin 2013) that, while using the least rectangles method (Alekseev 1962), it is possible to obtain a linear functional relation employing the same data on the river runoff and precipitation that once formed a curvilinear relation. That is, having a curvilinear correlation between river runoff depth and precipitation depth, reducing them to equally provided values, the curvilinear relation can be approximated in the form of a straight line, which can show the functional relation (Alekseev 1962). Thus, the linear correlation equation is applicable for any form of the "runoff-precipitation" graph.
Discussion and results
The development of the theory and methodology of calculation of water balance elements in the form of the linear correlation model has resulted in the high accuracy of calculation of water balance and its elements (Iofin 2013). For ease of comparison, the accuracy of calculation of the water balance parameters with the help of the existing methods, calculations according to the linear-correlation model and water balance calculation accuracy are presented in the tables below.
The values represented in the tables prove that errors can vary greatly whatever the size of the catchment area. It should be noted that serious errors in calculating water-balance parameters with the help of equation (1) vary according to the type of soil. The latter can determine gravitational moisture seepage to the groundwater level. In the areas with well-permeable soils the calculation error of water balance parameters using equation (1) will be big. For poorly permeable soils, the error will be small. It is very important that such serious errors a priori will result in overestimation of investments in water engineering.
Y
5
Table 1.
Accuracy of water balance parameters calculation.
№ River station F-catchment area, km2 Observation period, years Observation values Water balance according to P=Y+F„ mm Equation of regression Y = f(ñ Calculations according to linear-correlation equation
P, precipitation, mm Y, runoff, mm E, evaporation, mm Y, mm E, mm u, mm Balance, mm
1 2 3 4 5 6 7 8 9 10 11 12 13
Podmoskovnaya Water Balance Station
1 The Medvenka River - upstream of the mouth of the Zaksa River 21.5 63 674 193 340 533 y=0.55x-166 193 302 161 656
2 The Medvenka River - downstraem of the mouth of the Zaksa River 40 63 674 212 340 552 y=0.53x-146 212 276 181 669
3 The Lyzlovo Ravine 1.76 63 674 73 340 413 y=0.28x-107 73 382 201 656
4 The Kulibin Ravine 0.44 50 573 42 340 382 y=0.18x-61.8 42 344 198 584
5 The Lesnoy Ravine 0.066 50 573 33 340 373 y=0.29x-131 33 454 85 572
Catchment Areas of Cuban Rivers
6 The Chaparra-el Roble River 395 40 1063 188 606 794 Y=0.411P-249 188 524 270 982
7 The Gibarra-el Jobo I River 80.0 40 1127 292 658 950 Y=0.623P-410 292 628 179 1099
8 The Mayari-la Emajagua River 1060 40 1458 372 768 1140 Y=0.539P-414 372 786 317 1475
9 The S. de Tanamo-el Infierno River 325 40 1766 602 638 1240 Y=0.534P-341 602 561 534 1697
10 The Toa-el Torro II River 326 40 1688 700 594 1294 Y=0.64P-380 700 507 394 1601
11 The Toa-el Aguacate River 753 40 2439 1522 537 2059 Y=0.811P-456 1522 586 357 2465
12 The Jaguani-A.Prieto River 182 40 2758 1924 600 2524 Y=0.892P-536 1924 548 238 2710
Italian Rivers
13 The Mulargia River 64.8 42 542 161 300 461 Y=0.66P-198 161 300 82,8 544
14 The Candelaro River 85.4 42 611 74,2 394 468 Y=0.63P-281 74,2 394 145 613
15 The Celone River 2186 42 730 179 446 625 Y=0.34P-134 179 446 105 730
16 The Enza River 884 42 870 347 244 591 Y=0.55P-134 347 244 284 875
17 The Ofanto River 272 42 1000 350 418 768 Y=0.60P-251 350 418 234 1002
18 The Mannu River 488 35 502 122 172 294 Y=0.37P-63.6 122 172 207 501
Tabic 2.
Evaluation of accuracy of water balance calculation.
№ River station P, precipitation, mm Water balance according to P=Y+E, mm Calculation error according to P=Y+E , % Water balance according to linear-correlation model, P mm Calculation error according to linear-correlation model,%
Podmoskovnaya Water Balance Station
1 The Medvenka River - upstream of the mouth of the Zaksa River 674 533 21 656 3
2 The Medvenka River - downstraem of the mouth of the Zaksa River 674 552 18 669 1
3 The Lyzlovo Ravine 674 413 39 656 3
4 The Kulibin Ravine 573 382 33 584 2
5 The Lesnoy Ravine 573 373 35 572 0
Catchment Areas of Cuban Rivers
6 The Chaparra-el Roble River 1063 794 25 982 1
7 The Gibarra-cl Jobo I River 1127 950 16 1099 2
8 The Mayari-la Emajagua River 1458 1140 22 1475 1
9 The S. de Tanamo-el Infierno River 1766 1240 30 1697 4
10 The Toa-cl Torro II River 1688 1294 23 1601 5
11 The Toa-el Aguacato River 2439 2059 16 2465 1
12 The Jaguani-A.Prieto River 2758 2524 8 2710 2
Italian Rivers
13 The Mulargia River 542 461 15 544 0
14 The Candclaro River 611 468 23 613 0
15 The Celone River 730 625 14 730 0
16 The Enza River 870 591 32 875 0
17 The Ofanto River 1000 768 23 1002 0
18 The Mannu River 502 294 41 501 0
All the water balance parameters are calculated analytically with the help of the linear-correlation model. At the existing level of knowledge all the parameters cannot be calculated with the same accuracy with the help of other methods.
Conclusion
In this article, a new robust method of calculating water balance and water balance parameters has been developed. The solution proposed in this article is based on the linear correlation model, the genetic theory of runoff and the method of the least rectangles. The suggested approach is useful both for the theory of water balance and the accuracy of calculating water balance parameters. The presented approach has many advantages: the number of calculated water balance parameters and the accuracy of their calculation. The model proposed in this article can be recommended for practical use.
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