Hydrotechnical Construction, Vol. 32, No. 10, 1998 ECOLOGY

AND

HYDRAULIC

ENGINEERING

M O D E L I N G T H E O X Y G E N R E G I M E OF S T R A T I F I E D RESERVOIRS

A. G. Sokolov a n d I. L. D m i t r i e v a

The existence of stable thermal stratification in a reservoir during the summer alters appreciably the conditions under which its oxygen regime is formed, since the thermocline prevents diffusion of oxygen supplied to the body from the atmosphere and during photosynthesis of algae, and from the epilimnion to the hypolimnion. As a result, water supersaturated with oxygen may be observed in the epilimnion, and, conversely, an oxygen deficit in the hypolimnion. In the near-bottom layers, its content may be diminished virtually to zero with fatal consequences for near-bottom and bottom fauna. The thermal regime of a stratified reservoir can be schematized by a two-layer model consisting of an epilimnion and hypolimnion, which separates the plane with a pronounced change (jump) in water temperature, being conditionally adopted as the thermocline. Here, the temperature of the upper boundary of the thermocline can be assumed equal to or less than the temperature of the water surface, and the temperature of the lower boundary equal to or greater than the temperature of the water at the bottom, depending on specific conditions. In the latter case, the temperature variation of the water throughout the depth of the epilimnion and hypolimnion is assumed to be linear. A similar assumption is also made for the variation in the oxygen content over the depth of the hypolimnion, which, as a rule, decreases with depth in mesotrophic and eutrophic bodies of water. As for the epilimnion in which, in contrast to the hypolimnion, the oxygen content is largely determined not by diffusion, but by adequate transfer, i.e., by the mixing of water (runoff, wind-generated, and convective flows), its oxygen content may be assumed constant over the depth. A mathematical model [1] on which the hypothesis of the ideal mixing of water in a reservoir on the whole, or taken layer at a time, is based, can then be used to calculate the oxygen regime of a stratified reservoir. Using this model, the oxygen regime is calculated successively in the epilimnion at first, and then in the hypolimnion. In that case, the thermal regime of some layers is assumed to be known, and the average velocities of the flows in these layers are assumed to be directly proportional to their average water temperature [2]. Proceeding from this condition, which is valid for poor circulating and deep reservoirs, irrespective of the degree of circulation in the latter (for which temperature stratification in the summer is also characteristic) and the draw down of water into the tailrace is known, the flow of water is determined separately for the epilimnion and hypolimnion, which will also determine the water volume in these layers, and, accordingly, their oxygen regime. The fact that the oxygen-saturation concentration of water O~ in both layers is determined by the surface temperature of the water is common to calculation of the regime in the epilimnion and hypolimnion. This temperature controls the equilibrium concentration of O3 in the epilimnion. Maximum oxygen concentration in the hypolimnion obviously cannot exceed the equilibrium concentration in the epilimnion, since this would lead to diffusion of oxygen from the hypolimnion to the epilimnion, and its emission to the atmosphere. Conversely, the reaeration coefficients for the epilimnion and hypolimnion cannot be the same. For the case in question, these coefficients characterize the rate of oxygen supplied from the atmosphere to the epilimnion, and the rate of oxygen diffusion from the epilimnion to the hypolimnion, which will differ, since the diffusion of oxygen into the hypolimnion obstructs the thermocline to a greater degree than a more stable stratification, i.e., than a larger temperature jump in the layer of the thermocline. As we know, the average stay T of water (substance) in a reservoir under conditions of a steady-state regime for any time interval At can be determined from the equation [3] W T = ~-At,

(1)

where W and V~ are, respectively, the volume of the reservoir and the runoff of water from the latter during time At.

Translated from Gidrotekhnicheskoe Stroitel'stvo, No. 10, pp. 27-32, October, 1998. 618

0018-8220/98/3210-0618520.00

@1999 Kluwer Academic/Plenum Publishers

It is then possible to write (2)

u = l/T,

where u is the average velocity of the flow, and I is the length of the reservoir. If the volumes of the epilimnion We and hypolimnion Wh, and the average water temperature in each of these layers t~ and t~ are known, :~ to = teWe + t~Wh We+Wh '

(3)

where t ~ is the average-weighted water temperature in the overall volume of the epilimnion and hypolimnion. If the average velocities of the flow in the epilimnion ue and hypolimnion Uh are assumed to be directly proportional to the average water temperature of these layers t~ and t~, we obtain, proceeding from Eqs. (1)-(3) and after a series of transformations, t~

Te = ~ T ;

(4)

t~

Th = ~ T ;

(5)

v: -

(6)

T~

V~ - W h A t

Th

'

(7)

where Te, Th, V~e, and Vh are the average stays of the water (substance) and runoff during time At, respectively, in the epilimnion and hypolimnion. Let us write the differential equation of the first-order reaction, which characterizes the process of the water's saturation with atmospheric oxygen: dD

"dt = - k D ,

(8)

where D is the oxygen deficit, and k is the reaeration coefficient. According to Brekhovskikh [4], it is possible to write Eq. (8) in the following form: dD

F - ~KLD,

(9)

where F and W are, respectively, the area of the water surface and the volume of the reservoir, and KL is the mass-transfer coefficient. It follows from Eqs. (8) and (9) that KL = k W . F

(10)

According to [4], it is possible, on the other hand, to write KL = Din~5,

(11)

for KL, where Dm is the coefficient of the molecular diffusion of oxygen, ~ is the thickness of the film, diffusion through which controls the process of gas absorption by the liquid. A pronounced oxygen-concentration gradient from the value O~ on the interphase surface to a value characteristic of its entire liquid volume is observed in this film. The value of 5 is inversely proportional to the velocity of the flow. The film thickness ~ decreases, and the coefficient of mass transfer KL increases with increasing flow velocity. This is confirmed by results of laboratory experiments with more or less vigorous agitation of the water, and also by data derived from field investigations. The thickness 8 of the diffused layer increases on transition from rivers to reservoirs, where the thermal layer (warm and cold) on the water surface plays a special role. Its dynamics determine the dynamics of the diffused layer ~, and, consequently, the value of the mass-transfer coefficient KL.

619

Let us write Eq. (11) for the hypolimnion for two hypothetical conditions - with consideration and disregard of the overlaying layer (epilimnion):

g ~ = Dm/6; Kw: Dm/Sw,. =

(12) (13)

It is possible to write 6w = 5wh for each given computed time interval At (months), since the flow velocities for the entire W and partial volume Wh of the reservoir (for a runoff Vh) will be negligible, the difference between them nonexistent, and the computed temperature conditions on the surface of the water will be the same for any case. Using Eq. (10), it is then possible to write

wW

kU Wh Fh'

(14)

kW = khWhWhF FhW"

(15)

kW = khWhWh

(16)

kh

:

hence,

In the special case where Fh --~ F:

Fh"

Let us introduce the notation a -

WhF

WFh

or a = Wh/W.

(17)

= akU,

(is)

Then,

where kW is the aeration coefficient, which is averaged over time At and which is determined for the hypolimnion with consideration of the overlaying layer - epilimnion, kWh is the reaeration coefficient, which is averaged over time At and which is determined for the hypolimnion, disregarding the overlying layer - epilimnion - by selection, or by analytical means in accordance with the method outlined in [1]. The smaller the volume of the hypolimnion with respect to the entire volume of the reservoir and the greater the temperature stratification of the water, i.e., the lower the average water temperature of the hypolimnion with respect to the average temperature of the water of the entire reservoir, the lower the coefficient k W . The oxygen regime during the summer stratification is therefore calculated for the epilimnion and hypolimnion, just as for individual reservoirs, but with allowance for characteristic features of determination of the reaeration coefficient and saturation concentration for the hypolimnion. In the other months of the year, the calculation is performed for the reservoir on the whole, which is treated alternately as a continuous epilimnion at first, and then as a continuous hypolimnion. The background (averaged over the entire volume of the epilimnion or hypolimnion) content of oxygen is determined as a result of this calculation. In that case, the oxygen content is assumed to be constant over the depth in the epilimnion during the summer stratification, and decrease linearly with depth in the hypolimnion; this makes it possible to determine the oxygen content at the bottom from the equation O bet -- 2 0 h

-

-

03,

(19)

where O b~ is the oxygen concentration near the bottom, which is averaged over time At (month), and 0~ and 02h are the background oxygen concentrations averaged over time At in the epilimnion and hypolimnion, respectively. The oxygen concentration in the runoff from the reservoir for all months can be found from the equation

o2 = o v: + o v2

y:+y}

(20)

To illustrate the practical use of the mathematical model proposed, let us calculate the oxygen regime of a hypothetical stratified reservoir, the temperature, water, ice, hydrochemical, and hydrobiological regimes of which 620

T A B L E 1. Initial Data for Calculation of Oxygen Regime of Epilimnion

t~g

Wf

Month

I:~BOD" in 7

Wi/Wf,

tons

millions of m s 15.22 15.09 14.67 17.30 17.40 7.0/8.0 9.00 10.00

I 912 II 59 III 190 IV 192 V 155 VI 58.06 VII 65.69 VIII 60.39 IX 163 X 252 XI 39 XII 40 Total 1365.14 Average

17.4/17.9 18.00

18.45 17.95 14.71

Wireg, W[ eg, V~, En, t~vg/t~ur millions millions millions millions millions deg C of m s of m s of m s of m s of m a 15.66 -2.85 3.59 0.07 15.16 -2.621 3.65 0.09 14.88 -8.011 8.60 0.6 15.98 -12.91 7.78 2.94 17.35 -3.12 3.56 13.27/15.48 7.50 7.55 7.45 1.83 2.29 19.90 8.50 8.28 8.73 1.52 2.57 20.65 9.50 9.32 9.32 3.1 3.35 20.3 17.65 -2.58 3.47 15.53/17.13 17.95 -5.89 6.24 8.68 18.22 -3.23 3.84 3.36 18.20 -1.83 3.59 0.30 49.582 52.53 8.815/9.13 W,

Pr?~, Pg~-D),

T,

tons

tons

days

35 26 62 72 30 19 20.4 29.6 30 48 30 36 438

0 0 0 0 28 0 100.4 143.9 103.84 -275.8 0 0 100.34

170 162 57.6 37.1 172 122.7 173.1 92.3 205 94.5 169 308 108.29

Note: Wi, Wf, and W are, respectively, the initial, final, and average volumes of the epilimnion or hypolimnion, including those governed by the position of the thermocline from June through August; Wireg and W[ eg are, respectively, the initial and final volumes of the epilimnion, which are governed by flow regulation; t~g and ts~ r are, respectively, the water temperature averaged over the depth and the surface-water temperature of the epilimnion or hypolimnion; V~ and Vin are, respectively, the runoff and influx of water to the epilimnion or hypolimnion; Pi~ ~ is the supply of readily oxidizable organic substance (BODs) in the epilimnion or hypolimnion, respectively; Pr~ is the supply of dissolved oxygen to the epilimnion or hypolimnion from the river; P(~2 D) is the supply (loss) of oxygen in the epilimnion or hypolimnion due to pure photosynthesis (pure destruction) of phytoplankton; and, T is the stay of the water (substance) in the epilimnion or hypolimnion. can be schematized on the basis of corresponding regimes of the water bodies associated with the Zagorsk waterstorage power plant, which were observed in 1989 [1, 3, 5]. Of these regimes, let us, for this purpose, vary only the temperature regime, imparting a stratification to it in the summer (June-August). Let us assume that temperature stratification of the reservoir begins to develop in May and is completed toward the end of this month by the formation of a thermocline, which exists in the period from June through August, and is broken down in September. Here, the thermocline - a conditional plane that separates the epilimnion from the hypolimnion during the period from June through August - gradually sinks to ever greater depths, changing the ratio between their volumes, i.e., increasing the volume of the epilimnion, and, accordingly, reducing the volume of the hypolimnion. The dynamics of these volumes during the period from June through August (their initial, final, and average values) for each of the indicated months is presented in Tables 1 and 2. It should be noted that although these volumes are linked to the initial volumes of the water bodies of the water-storage power plant, their dynamics is, in contrast to the latter, caused not by flow regulation, but by a change in the position of the thermocline. In addition to the temperature regime of the epilimnion and hypolimnion, the dynamics of the volumes of the latter is, like the runoffs V~e and Vh , as well as the stay of the water (substance) in these bodies, basic to calculation of this element of their water balance from Eqs. (4)-(7) (Tables 1 and 2). Initial data and computational results for other elements of the water balance of the epilimnion and hypolimnion are also presented in Tables 1 and 2. For the case in question, the inflow of water to the hypolimnion is assumed equal to its runoff, i.e., Vin h = V~. The inflow of water to the epilimnion is determined from the equation

V ~ = V: + Vev + A W, where Vev is the volume of water evaporated from the surface of the water bodies of the water-storage power plant during a given month, and A W is the change in the total volume of the water bodies of the water-storage power plant for the month in question.

621

TABLE 2. Initial Data for Calculation of Oxygen Regime of Hypolimnion

Month

pBOD, " in

tons I II III IV V Vl VII VIII IX X XI XII

Total ....Average

91 59 190 192 155 223.22 124.35 69.17 163 252 39 40 1597.74

t~

Wt

wdw. millions of m s 16.22 15.09 14.67 17.30 17.40 10.4/9.3 8.75 7.40 17.4/17.9 18.00 18.45 17.95

W,

millions of m3 15.66 15.16 14.88 15.98 17.35 9.85 9.02 8.08 17.65 17.95 18.22 18.20

0 o t.,Jt..,

Vr,

millions of m s 3.59 3.65 8.60 7.78 3.56 1.21 0.70 1.07 3.47 6.24 3.84 3.59 47.3

millions of m 3

2.85 2.621 8.011 12.91 3.12 1.21 0.70 1.0 2.58 5.89 3.23 1.83 46.022

14.85

deg C 0.07 0.09 0.6 2.94 13.27/15.48 10.0/19.9 9.0/20.65 8.0/20.33 15.53/17.13 8.68 3.36 0.30 5-99/9.13

r.w ~ pO2 tons

02 P(P-D)' tons

35 26 62 72 30 10 5.6 9.5 30 48 30 36 394.1

0 0 0 0 28 -100.5 0 0 103.84 -275.8 0 0 -244.46

T,

days 170 162 57.6 37.1 172 244.2 397.2 234.6 205 94.5 169 308 117.62

Initial and final volumes of the epilimnion for each month ( June through August), which are associated with runoff regulation, are determined from the equations Wf = We +

AW

AW = we

w=w?-wL The readily oxidizable organic material (BODs) and dissolved oxygen supplied to the epilimnion and hypolimnion from river and drainage water is then calculated with respect to the volumes of inflows to the epilimnion and hypolimnion. The following approach was used to calculated the supply of readily dissolvable organic matter (BODs) associated with the destruction of phytoplankton, and also to evaluate the supply or loss of dissolved oxygen in the epilimnion and hypolimnion, which are governed, respectively, by positive and negative values of pure photosynthesis of phytoplankton. It was assumed that the oxygen produced during the photosynthesis of phytoplankton is intermixed, i.e., is subjected to adequate transfer only in the epilimnion, partially going over into the atmosphere on supersaturation, and its supply in the hypolimnion, like atmospheric oxygen, occurs only as a result of diffusion (turbulent or molecular), which is characterized by the reaeration coefficient. Proceeding from this, positive values of pure photosynthesis (pure production) were applied only to the epilimnion. Values of phytoplankton destruction were applied only to the hypolimnion, since decomposition of hydrobionts after their death takes place primarily in the near-bottom layers of water and the bottom sediments. Negative values of pure photosynthesis (pure destruction) were applied in an equal manner only to the hypolimnion (Tables 1 and 2). Using Tables 1 and 2, the mathematical model described in [1] and Eq. (18), let us calculate the oxygen regime in the epilimnion and hypolimnion separately. Here, let us also use several numerical values of the parameters of the indicated model, which are determined for the water bodies associated with the Zagorsk water-storage power plant [1]. The coefficient a in Eq. (18) can be found from Table 1 for the period of stable stratification (June through August), disregarding the quantities F and F h in Eq. (17): avl = 0.432; avIi = 0.485; and, aVHI = 0.54. The oxygen regime can be calculated sequentially - first for the epilimnion, and then for the hypolimnion. In the first case, the entire volume of the reservoir can be calculated for all months, with the exception of June through August, as a continuous epilimnion, and in the second case, as a continuous hypolimnion. The water temperature for all months, with the exception of May through September, can be assumed the same throughout the depth of the reservoir, and can be separated into surface temperature and temperature averaged over the depth in May and September, considering that the thermocline is formed and destroyed in these months. The effect of the surface temperature of the water on the saturation concentration, and, consequently, also on the background (averaged throughout the depth of the reservoir) concentration of oxygen dissolved in the water are considered in turn. In calculating the background oxygen concentration in the hypolimnion during the summer months (June

622

T A B L E 3. C o m p u t a t i o n a l Results of Oxygen Regime in Stratified Reservoir, m g / i i t e r With water withdrawn only from With water withdrawn from zones of zone of epilimnion thermocline, epilimnion, and hypolimnion Month epilimnion hypolimnion epilimnion 02 O hOt 02 O 2D~ D D 02 02 D 2 D 9 3.75 10.84 4.02 9 10.57 -II 3.75 10.84 4.05 10.549 - 6.82 7.76 7.42 7.16 6.80 7.21 7.78 7.37 III 8.08 6.28 8.96 5.39 -8.05 5.47 6.30 8.89 IV 2.44 11.06 2.68 10.82 -2.45 11.05 2.64 10.86 V 1.33 8.72 1.33 8.71 -1.33 8.72 1.33 8.71 VI 1.41 7.79 5.17 4.03 0.27 1.33 7.87 5.04 4.17 0.47 VII 1.36 7.71 4.42 4.65 1.59 1.25 7.82 4.34 4.73 1.64 VIII 1.12 8.00 4.59 4.54 1.08 1.05 8.08 4.19 4.93 1.78 IX 1.31 8.40 1.97 7.74 -1.26 7.78 8.46 1.93 X 3.48 8.21 3.90 7.78 -3.66 8.06 8.03 3.63 XI 1.92 11.43 2.08 11.28 -1.96 -11.39 2.02 11.33 XII 1.86 12.64 2.01 12.48 -1.87 12.63 1.99 12.51 h y p o l i m n i o n

9

- -

9

- -

With water withdrawn only from zone of hypolimnion epilimnion hypotimnion D 02 D 02 0 2b~ 3.64 10.96 4.02 10.57 -9 6.58 8.00 7.38 7.2 -7.73 6.63 8.89 5.46 -2.35 11.15 2.66 10.84 -9 1.32 8.72 1.33 8.7 -1.19 8.01 4.69 4.51 1.01 0.97 8.10 4.16 4.91 1.72 0.76 8.36 2.99 6.14 3.92 1.06 8.65 1.88 7.83 -3.42 8.27 3.88 7.81 -. 1.88 11.48 2.08 11.27 -1.80 12.70 2.01 12.49 --

Note: D) oxygen deficit in epilimnion or hypolimnion; 02) background (averaged over depth) oxygen content in epilimnion or hypolimnion; O b~ oxygen content near b o t t o m of reservoir.

through August), the surface water t e m p e r a t u r e and t e m p e r a t u r e averaged over the depth of the hypolimnion are also considered. As for the epilimnion, the surface water t e m p e r a t u r e and t e m p e r a t u r e averaged over the depth are assumed to the same in these months. In these cases, i.e., when unstable or stable stratification exists, therefore, the oxygen regime of the hypolimnion, or the reservoir on the whole, is calculated with consideration of two factors: the surface water t e m p e r a t u r e and the water t e m p e r a t u r e averaged throughout the depth. Results of calculation of the oxygen regime of a hypothetically stratified reservoir in accordance with one of the methods based on the m a t h e m a t i c a l model outlined in [1], the so-called basic method with consideration given to the latter's modification, are presented in Table 3. As is a p p a r e n t from this table, the oxygen content near the b o t t o m of the reservoir for its prescribed t e m p e r a t u r e stratification during the s u m m e r was lower t h a n the ecological standard of 4 m g / l i t e r during all months of this period with a minimum of 0.47 m g / l i t e r in June (with a water intake from the zone of the thermocline, epilimnion, and hypolimnion). T h e latter is caused by high negative pure photosynthesis (pure destruction) of phytoplankton during this month, which amounts to minus 100.5 tons in oxygen units (Table 2). Nevertheless, results of the modeling of flows in stratified bodies of water [6] indicate t h a t a phenomenon t h a t has acquired the n a m e selective water intake occurs when water is taken from them: water enters an intake preferentially from layers situated at the level of the opening by virtue of Archimedes forces, depending on the stability of the stratification, water from the overlying and underlying layers is either taken in negligible amounts, or none whatsoever enters the opening. In the latter case, the thickness of the intake zone will be smaller t h a n the total depth of the water body, and encompass only the epilimnion or hypolimnion, depending on what level the intake opening is located. As Bocharov and Vasil'ev demonstrate [6], with these openings positioned in the zone of the thermocline, the velocities of the flows in the epilimnion and hypolimnion would be directly proportional (inversely proportional), depending on the t e m p e r a t u r e (density) of their water. A similar relationship is also used in the proposed mathematical model for the oxygen regime of stratified reservoirs. In reservoirs for hydroelectric power plants, the intake openings of the d a m are situated, as we know, at the level of the stagnant volume. If it is assumed t h a t this level falls within the zone of the thermocline of near it, use of this relationship can be considered feasible. If, however, water is withdrawn only from the hypolimnion ( b o t t o m openings) or epilimnion (spillway d a m ,or selective intake), use of this relationship cannot be recognized as feasible. To ascertain as to what degree selective intake from the upper or lower layers of a reservoir will affect the oxygen regime of the epilimnion and hyDolimnion, and, a m o n g other things, the oxygen content near the b o t t o m , let us calculate this regime for different conditions using the proposed m a t h e m a t i c a l model. Let us examine two extreme cases. In the first case, the intake of water V~e is accomplished only from the epilimnion; here, the runoff from the hypolimnion Vh - 0. In the second case, conversely, the intake of water Vh is carried out only from the hypolimnion; here, the runoff of water from the epilimnion V~e = 0. Determining the elements of the water balance separately for

623

the epilimnion and hypolimnion by a procedure similar to that described above for these two cases, the oxygen regime in the epilimnion and hypolimnion can be calculated by the basic method, referring the destruction of phytoplankton here only to the hypolimnion, and the positive and negative values of pure photosynthesis of phytoplankton only to the epilimnion and hypolimnion, respectively. The results of this calculation are presented in Table 3. Table 3 indicates that as would be expected, the lowest oxygen content near the bottom is obtained when water is withdrawn only from the epilimnion, and the highest content when water is drawn only from the hypolimnion. It can be seen from the three alternate schemes cited in this table for the withdrawal of water from a reservoir that the oxygen content increases in both the epihmnion, and also the hypolimnion as the depth of selective intake increases. The first is governed by increasing importance of the role of photosynthetic aeration in the formulation of the oxygen regime of the epilimnion as the rate of water exchange decreases in the latter. The second is dictated by the increasing role of turbulent diffusion and adequate oxygen transfer to the hypolimnion as the rate of water exchange in the latter increases. In effect, the tendency of the oxygen content to vary in these layers with varying depth of selective water intake will not be so unique as computational results would indicate. The fact is that for this calculation, variation of the balance between the production and destruction of phytoplankton in the reservoir, which is associated with variation in the depth of selective intake, was disregarded in all three alternate schemes of the withdrawal of water from the reservoir. Consideration of this factor, and also the variation in the temperature regime of the reservoir in connection with variation in the depth of selective water intake is a rather complex problem; it can be solved, however, by means of alternate calculations to provide a quantitative estimate of their possible effect on the oxygen regime of the reservoir, and in this manner, to substantiate the effectiveness of the use of selective water intake as a nature-conserving measure.

Conclusions A mathematical model is proposed for the oxygen regime of a stratified reservoir. A model is based on representation of the reservoir on the whole or its layers individually as an ideal mixer. The thermal regime of stratified reservoir is schematized by a two-layer model - epilimnion and hypolimnion, which provide the plane with a pronounced variation in water temperature (thermocline). In calculating the oxygen regime, the epilimnion and hypolimnion are considered individual reservoirs with their own water balance, which is linked, however, to the water balance of the reservoir on the whole. Characteristic features of the aeration and reaeration of the epilimnion and hypolimnion, and also the selectivity of water intake from the reservoir are considered here. Practical use of the model is illustrated in an example of the calculation of the oxygen regime of a hypothetical stratified reservoir, the characteristics of which are determined on the basis of field data derived from observations on the water bodies of the Zagorsk water-storage power plant. The proposed model can be used to predict the oxygen regime of stratified reservoirs and the ecological feasibility of nature-conserving measures directed toward improving the quality of their water and conserving fauna.

REFERENCES .

2. 3. 4. 5. 6.

624

A. G. Sokolov and I. L. Dmitrieva, "Dynamic model of the oxygen demand of a reservoir," Gidrotekh. Stroit., No. 6 (1996). K. I. Rossinskii, Thermal Regimes of Reservoirs [in Russian], Gidrometeoizdat, Leningrad (1981). A. G. Sokolov, "Use of a model of an ideal mixer to calculate the hydrochemical quality indicators of reservoir water," Gidrotekh. Stroit., No. 4 (1996). V. F. Brekhovskikh, Hydrophysical Factors of Oxygen-Regime Formulation for Reservoirs [in Russian], Nauka, Moscow (1988). A. G. Sokolov, "A mathematical model of the primary production and destruction of phytoplankton in active reservoirs and reservoirs under design," Gidrotekh. Stroit., No. 5 (1996). O. B. Bocharov and O. F. Vasil'ev, "Semi-empirical model of selective withdrawal from a stratified water body," Vodn. Resur., No. 3 (1982).