Probabilistic model of water reservoir for Southern India conditions

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Probabilistic model of water reservoir for Southern India conditions

L. Elango

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 VlsVend = Vds + Vf, S = 0, U = Upl, D = 0. (3)

Vf 0Vend = Vds, S = 0, U = UplVf , 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)

нereY - 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=KЧXЧF, (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 = (KF + ) 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.

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.

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: 0ч0 and Vls ч 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 = 0ч0 > Middle V = V1 = 0; j = 2 > V = 0ч12? Middle V = V2=6; j = 3 > V = 12ч24? Middle V = V3 =18; j = 4 > V = 24ч36> Middle V = V4 = 30; j = 5 > V = 36ч36? 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?

Conclusions and problems

water reservoir balance

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?

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