# A method for determining areas as a function of depth

When it is desirable to calculate what area of a hydroelectric reservoir occupies the strip whose depth ranges from zero to a given depth, for example the strip with depth ranging from zero to 20m, you can use the method here presented called the inverted pyramid. This method uses as a reservoir model the “inverted pyramid” of the figure, and it is necessary to know the size of surface area A and dam depth H. Although the model is very simple, there has been reasonable agreement with known areas, determined by topography.

Assuming the water surface area to be **A** when the dam depth is **H** is is possible to calculate the area **a** that the water surface (colored in the figure) would have for depth **h**. It is easy to show that area **a** is proportional to the square of depth **h**, in other words

**a** = c x **h**^{2},

where c is the proportionality constant. This constant is:

c = **A**/**H**^{2}

In the following lines, by specifying **z** as the depth limit until where there is bubble emission, the area **a** will be identified as the area that does not emit bubbles.

After it is possible to calculate, by difference, the area **S** that emits bubbles, bounded by the shore and depth **z**. From bubble emission measurements maximum depth **z** is determined until where bubble emissions occur and **h** is calculated being **h**=**H**–**z**. The area **S** which emits bubbles with then be **S**=**A**–**a**, or

**S** = ** A** – (**A**/**H**^{2})x**h**^{2}.

Application of this method to two reservoirs:

1 – Tres Marias reservoir (on the São Francisco river, **A**=1009,32(km)^{2}, **H**=50,2m). Assuming there is bubble emission from the shore to a depth of **z**= 23,3m, **h**= 26,9m is found and the bubble emitting area **S**=719,50(km)^{2} is calculated, to be compared with the known area, calculated from map contour lines, of 693,57(km)^{2}. An overestimatimation of 3,7% is noted.

2- Tucuruí reservoir(on the Tocantins river, **A**=2430(km)^{2}, **H**=65,5m). Assuming there is bubble emission from the shore to a depth of **z**=20,4m, **h**= 45,1m is found and the bubble emitting area **S**=1278(km)^{2}, to be compared with the known area, calculated from map contour lines, of 1502,35(km)^{2}. An underestimatimation of 15% is noted.