УДК 551.48.
PROBABILISTIC MODEL OF WATER RESERVOIR FOR SOUTHERN
INDIA CONDITIONS
L. Elango
Anna University, Chennai, India
V.V. Ilinich
Moscow State University of Environmental Engineering, Russiа
(Publication was prepared due to support of Indo-Russian International Long Term Project)
Today human using of Southern India (in particular Tamil territory) supposes the
increasing of water demands for successful development of national economy. However, water
resources are limited and therefore importance of their rational using is rising with point of view
both finance profit and environmental defense. In such conditions a big contribution may be
obtained from creation and using of artificial water objects: water reservoirs, tanks, ponds.
Tamil territory has water reservoirs, but there are not effective operative rules of management
by that. It is not possible to manage successfully by water reservoir without probabilistic model
of that’s work because operative rules of management have to be based on probabilistic
evaluations of water balance components for each future time interval. So, necessary of such
model is objective today reality.
The Madurantakam tank was choused for research. The
tank is located on Tamil territory within Palar river basin, we can say the tank have not any
strongly expressed differences comparatively other water reservoirs of the territory and may be
considerate enough represented for future total conclusions. So, subject of this research is water
reservoir –Maduramtakan.
Main purpose of research is creation of probabilistic model for water reservoir’s storage.
The purpose requires deciding next problems:
тo analyze components of water balance for the water reservoir enough correctively in
limit conditions of observation data and to define main statistical characteristics for the
components;
тo create probabilistic model in form of totality of month probabilistic functions for
water reservoir storage.
Water balance of water reservoir for the month is defined by next equation
Vend = Vst + W + X        Ew    U  S,
(1)
where Vend - volume of water in reservoir to the end of month; Vst - volume of water in reservoir
to the start of month; W - volume of runoff to the water reservoir during month; X - depth of
month precipitation on the surface of water reservoir;  - the area of water reservoir;  - depth
of filtration losses through bed of water reservoir; Ew - depth of evaporation from surface area
of water reservoir; U - volume water of consumption; S - volume, which outputs water
reservoir without use for consumers.
Notice, that U is real volume water of consumption, which can be less plan water of
consumption (Upl) in the case deficit of water (D). According to simplest normal rules of water
reservoir exploitation - real volume of water reservoir must not exceed Normal reservoir level
and not be less Dead storage level. Volume between Normal reservoir level and Dead storage
level is named Live storage (Vls). Dead storage (Vds) is located below Dead storage level.
Together they compose Full storage: Vfl = Vds + Vds. For the using water balance equation (1) we
must introduce term «fictive volume» (Vf), which is not limited volume (  Vf  ). Next
cases can take place during management by water reservoir according to usual normal rules:
Vf  Vls
 Vend = Vfl ,
S = Vf  Vls, U = Upl,
D = 0.
(2)
0  Vf  Vls  Vend = Vds + Vf, S = 0,
U = Upl,
D = 0.
(3)
Vf 0
 Vend = Vds,
S = 0,
U = UplVf , D = Vf .
(4)
Precipitation. Meteorological station Minambakkam is located near river basin Palar and
have enough represented time series of observation for the month precipitation – 28 years.
According to the observation data were defined probabilistic functions of month precipitation
X(P) [3].
Runoff. Runoff to the water reservoir is defined from water balance equation (5)
W = Y  F ; Y = X   Es,
(5)
нere Y - depth of runoff;  - depth of loses due to interception and filtration through soil;
Es – depth of ewapotranspiration from soil surface area of watershed.
Loses shares from rainfall precipitation due to filtration were researched on the base of
long duration observations within Palar river basin by researchers of Anna University (Chennai,
India) and represented in scientific reports and publication [1]. Most difficulties there were for
definition of evapotranspiration (Es) because it’s data of observations absent. Approximate
calculations were made by United Department of Technical Co-operation for Development in
technical report [2] on the base Penman formula, and dependence between precipitation and
evapotranspiration were analyzed. On the base this analysis we could take coefficient Ke = 0.4,
which expresses approximate share of month evapotranspiration from precipitation. However
we must take into consideration when month precipitations are not more 4 mm, all water is
discharged for start interception and than for evapotranspiration. [1]. So, runoff (W) in equation
(2) we can represent in next view
W=K×X×F,
(6)
K – complex mathematic operator, which satisfies conditions of [1] and [2].
Surface water evaporation. That component of balance equation (1) for water reservoir is
enough considerable for the Southern India because solar radiation and temperature are very
high. Data of observation on the meteorological station Minambakkam were used. The station
has enough represented time series of observation for the month evaporation from surface of
water – 24 years (1959–1982 гг.). It is need notice, that 12 years observations were made
together both evaporation and precipitation (1971-1982 гг.). Analysis of the together
observation data showed: a depth of evaporation from water surface much more precipitation
depth during same month (in 100 times and more in several cases). Average of annual
evaporation depth is equal 2027 mm, but of annual precipitation 1133 mm. Approximately,
similar rations (precipitation – evaporation) there are on other nearest meteorological stations.
For composition of right water balance there are necessary to evaluate dependence
between precipitation and surface water evaporation for every month. Usually, if big
precipitation have take, then evaporation is small. It is explained by existing of clouds and
consequently by small solar radiation and temperature. However trends of dependences may be
differed for different regions and time. Dependences for water balance composition were taken
from analysis of appearance of probabilities both month precipitation and evaporations during
same years [3]. On the base that analysis we can conclude that is permissibly to take probability
of evaporation Pe = 1 – Px. Such combination gives unfavorable variant both for water
supplying (small precipitation and big evaporation) and for protection from storm runoff (big
precipitation and small evaporation). We have added guaranties, which must have take place
because of different systematic mistakes.
Filtration losses through bed of water reservoir. Careful data about filtration losses
through bed of water reservoir are absent today for Madurantakam tank, therefore today we
can’t use that member of equation (1), however that component must be defined in future.
Consumption. For the case of agriculture using, usually, values of water consumption
depends on from demands of water by croups during vegetative period and soil moisture, which
depends on precipitation during that period and before. However, rice requires added irrigation
during rainfall down even, because there is a necessary to support concrete level on the rice
fields. Therefore, at first, the plan water requirements to the water reservoir must be defined by
dependence on regime of croup demands and second by on regime of precipitation. At last,
rational regime of consumption may be defined from model of water reservoir working. The
first iteration of plan consumption for deciding water balance equation (1) was chosen on the
base of analysis of real demands during of water reservoir working and analysis of dependence
between real demands and precipitation during previous period the first iteration of plan
consumption for deciding water balance equation (1) was chosen. So month potential water
resources (Z) for water reservoir we can express with help equation:
Z = W + X    Ew   , or Z = (KF + )  X  Ew  .
(7)
Consequently probabilistic functions of month potential water resources Z(P) are defined
according to: probabilistic precipitation functions, probabilistic curves of surface water
evaporation and dependence between probabilities of surface water evaporation and
precipitation (Pe = 1 – Px). Notice that functions of month potential water resources Z(P) can be
negative (mark minus) when the losses (Ew  ) exceed runoff and precipitation
(Ew    W + X   ), in particular during dry month (when probability of exceeding of values
Z is big). Probabilistic functions of month potential water resources Z(P) for different seasons
are represented on graph 1.
Z[mln cub m]
Graph 1. Probabilistic curves Z(P)
40
35
30
25
20
15
10
5
0
-5 0
-10
Apr
Jul
Nov
2 линейный
фильтр (Nov)
0,2
0,4
0,6
Probability P
0,8
1
2 линейный
фильтр (Jul)
2 линейный
фильтр (Apr)
According the logical formalization (2)…(4) of most simple rules of water reservoir
exploitation, probabilistic functions of month potential water resources Z(P) and means of plan
consumption were defined conditional probabilistic curves of storage volume to end of each
month in depended on conditions of five conditional means of start volume, which were taken:
1) i = 1 →Vst = V1 =0;
2) i = 2→Vst = V2 =6;
3) i = 3→Vst = V3 =18;
(8)
4) i = 4→Vst = V4 = 30;
5) i = 5→Vst = V5 = 36.
For each value probability of month curve Z(P) were defined values
Vend = Vst + Z(P) – U according to logical conditions (2)…(4). Values S did not calculate since
variant of storm flood did not take to consideration. So the obtained conditional probabilistic
curves of volume storage Vcon(P), represented by different kind of points on graph 2.
Graph 2. Points of Conditionall probabilistic curves
of water storage Vcon(P) - August
V1+Z-q
V2+Z-q
Volume [mln cub m]
V3+Z-q
V4+Z-q
36
32
28
24
20
16
12
8
4
0
V5+z-q
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
P
For obtaining full probabilistic functions of month storage volume was used formula of
full probability for every choused interval (i) of storage volume. Consequently, preliminary full
diapason of change storage volume (0 - Vls) is divided on the several intervals (j). In the most
simple imagine for our case the formula of full probability is represented
dPm,j = ∑{dPconm,j, i [Vm,j,i(Vm-1,i)]  dP(Vm-1,i)},
(9)
dPm,j - full probability for every interval (j) of storage volume; dP(Vm-1,i) - probability of storage
volume to start of the month; dPconm,j, i [Vm,j,i(Vm-1,i)] - conditional probability of storage
volume to end of the month; m
- numbers of month; j
- numbers of taken
intervals for storage volume and probability changes; i - numbers of conditions for the start
month storage volume;
Vm-1,i - storage volume to start of the month – Vst, which corresponds to storage volume to end
of the previous month (m-1) and to condition (i); Vm,j,i - storage volume to end of the month for
condition (i) of storage volume to start of the month (Vst)-Vm,j,i= Vst + Z – U according to logical
formalization (2)…(4).
Probability of conditional probabilistic curves Vcond(P) is changed within the volume
storage intervals. Therefore first interval and last interval are represented refers to storage
volume: 0÷0 and Vls ÷ Vls since change of probability take place on conditional probabilistic
curves (graph 2). Usually, 5 – 8 intervals are enough for exact deciding of task. In our example
were taken 5 intervals:
j = 1 → V = 0÷0
→ Middle V = V1 = 0; j = 2 → V = 0÷12→
Middle V = V2=6; j = 3 → V = 12÷24→ Middle V = V3 =18; j = 4 → V = 24÷36
→
Middle V = V4 = 30; j = 5 → V = 36÷36→ Middle V = V5 = 36.
It is necessary to take to consideration that values Vst (Vsti=1=V1=0;
Vst i = 2 = V2 = 6; Vst i = 3 =V3 =18; Vst i = 4 = V4 = 30; Vst i =5 = V5 = 36) are equal to middle
values of taken intervals ( j = 1….. j = 5). We can say that each interval is characterized by
consequent value Vst. Therefore i = j. It is essential position for use formula (9) for definition
values Pm,j. Means of dP (Vm-1,i) – probabilities of storage volume to start of the first month are
unknown values, therefore the means are taken any for first iteration of calculation, but summa
of their must be equal 1 (∑dP(Vm-1,i) = 1). For each next month means of dP(Vm-1,i) are taken
equal to means of Pm-1,j - full probabilities for every interval (j) of storage volume from
previous estimated month. After 12 month calculations the first iteration is finished and began
second iteration. Such estimations are finished fully when full probabilities of adjacent iterations
for every month and every interval (j) of storage volume will not be differ. Then last means of
full probabilities for each interval (j) of storage volume are base for the obtaining of total
probabilistic function of storage volume for every month. Formula (9) is realized by multiply of
matrix values of probabilities for storage volume to start of the month - dP(Vm-1,i) and
conditional probabilities of storage volume to end of the month - dPconm,j, i [Vm,j,i(Vm-1,i)].
Full probabilistic curves of water storage for Maduramtakam Tank to end February and
April month are represented on the graph 3. We can see from the graph that during April the
plan consumption is not provided even if we have maximum potential month water resources of
rare probability (P=0,05…0,02).
So, consumption is provided by ground water stories,
which can be reserved during wet period with help water reservoir. Totality of month
probabilistic functions (curves on the graph 3) is water reservoir probabilistic model, which is
made on the base simple Marcov chain because probability of volume storage of each month is
depended on probability previous month only. Obtained probabilistic model of water reservoir
can be used for the working out of operative profitable rules of management by consumption
and level during year. In that case a reverse task [4] must be decided relatively formula of full
probability (9). Then we can obtain answer to the question for example: What storage to start
month and consumption during month must take place for small probability of ecological
emergency to end of month?
V [mln cub m]
Graph 3. Probabilistic curves of water reservoir
storage to end month V(P)
36
32
28
24
20
16
12
8
4
0
Febr
April
0
0,2
0,4
0,6
0,8
1
P
Conclusions and problems
1. Analyze components of water balance for the water reservoir have showed that year
value of surface water evaporation is very big.
2. During dry years consumption within Paler river basin uses ground water stories in
spite of existing of water reservoir, which creates added surface water evaporation
approximately equal surface runoff.
3. We are need the working out of methods for decreasing added losses from water
reservoirs and realization that. For example, the multi-tier plants may be placed around water
reservoirs for decreasing of wind velocity and consequently evaporation.
4. We must consider possibilities for transformation of water stories from water reservoir
to ground water and creation underground water reservoirs.
5. Obtained probabilistic model of water reservoir can be used for the working out of
operative profitable rules of management by consumption and level during year. In that case a
reverse task [4] must be decided relatively formula of full probability (10). Then we can obtain
answer to the question for example: What storage to start month and consumption during month
must take place for small probability of ecological emergency to end of month?
References
1. M. Senhil Kumar and L. Elango. «Numerical Simulation of Groundwater flow Regime in a
Part of The Lower Palar River Basin, Southern India» (in the book «Modelling
Hydrogeology»).
2. Technical report: «Hydrogeological and artificial recharge studies, Madras» prepared by
Unated Nations Department of Technical Co-operation for Development (New York,
1987).
3. Ilinitch V.V. Probabilistic approach for modeling hydrological cathments. Anna
University, Chennai 2004.
4. 4. Ilinitch V., Perminov A. Management technology by multi-purpose water reservoirs for
protection of territory from floods. Scientific Conference: «Natural and technological
problems of protection and development of agricultural and forest environment». Book 2.
P. 17-22.