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Complex networks

The order of scarcity

Patterns discovered in a network of thousands of reservoirs in Ceará may help when it comes to facing drought and floods

ILLUSTRATION GABRIEL BITARAn international team of physicists and hydrologists were surprised when they analyzed how water flows along the rivers and streams that connect the almost 4,000 dams or reservoirs in the catchment area of the Upper Jaguaribe River, in southwest Ceará. They discovered that, although the majority of the reservoirs were constructed without considering any regional planning, together they form a network that is far from random. On the contrary, the network seems to be organized in such a way that the water is relatively well-collected and distributed throughout the region. “Country folk, even though they took local decisions without looking at the whole, constructed a system that is very close to the one that would have been constructed if it had been planned to be great,” says hydraulic engineer José Carlos de Araújo from the Federal University of Ceará (UFC), one of the authors of the study published in April on the website of the Proceedings of the National Academy of Sciences (PNAS).

Various properties of this network of reservoirs, especially the frequency with which they overflow one after another in rainy periods, obey the probabilistic patterns that are well-known to physicists who study complex networks, like electricity grids, neurons in the brain and the Internet. By exploring these mathematical regularities it would be possible to intervene in the network to make it more efficient and less vulnerable to droughts, floods or the bursting of dams.

In the drought polygon, Ceará suffers the whole year long from very dry weather, which is only interrupted by the rainy season, from February to May. Average annual evaporation exceeds precipitation by three or four times, meaning that the majority of the water courses are short-lived. To complicate the situation, most of the territory is of impermeable, crystalline rock covered by a thin soil, which contains hardly any underground water.

All rainwater would flow straight into the sea if it were not for more than 30,000 dams spread throughout the state, which form one of the densest networks of reservoirs in the world, with an average of one reservoir for every 6 sq. km. “Over the years there was uncontrolled building of these dams in order to give the people a certain degree of security as far as water is concerned,” explains hydrologist George Leite Mamede, from the University of Integração Internacional da Lusofonia Afro-Brasileira (Unilab), the main author of the PNAS article.

DNOCS ARCHIVEThe Orós Reservoir, the second largest in Ceará, in the Jaguaribe River basinDNOCS ARCHIVE

Researchers analyzed part of the basin of the Jaguaribe, the principal river in Ceará, which runs from its headwaters in the southwest of the state, on the border with Piauí, to where the river flows into the second biggest reservoir in Ceará, the Orós. The so-called Upper Jaguaribe basin covers an area of 25,000 sq. km, where 500,000 people live, mainly raising cattle and working in agriculture. In addition to Orós, which is capable of holding almost 2 billion cubic meters of water, the basin also has a further 17 strategic reservoirs with capacity for over 1 million cubic meters, which guarantee a supply of water even during the periods of drought that can continue up to three years. These strategic reservoirs are monitored by technicians from the Water Resource Management Company (Cogerh) of Ceará.

Most of the almost 4,000 reservoirs in the Upper Jaguaribe, however, are small constructions, holding volumes of less than 100,000 cubic meters, and made by ranchers and small farmers, sometimes with the support of the local town administration, but without any investigation into the impact the work might create. “Moreover, they’re not used rationally,” explains Araújo. “There’s no management system for the small reservoirs.”

In fact, this profusion of small reservoirs is not looked on kindly by most of the state’s managers of water resources. First, because they are unable to supply the population for the whole of the dry period, because they hold so little water and lose a lot through evaporation and infiltration. Then, because they are generally up-stream from the major strategic reservoirs, small reservoirs retain the water that would reach the big ones where its use is better controlled.

The small ones’ turn
But the Hydro-sedimentological Research Group of the Semiarid Region from the UFC, coordinated by Araújo, has been collecting evidence that small reservoirs do have their positive aspects. The most obvious one is the more egalitarian and economic spatial distribution of water resources throughout the region. If all the water ran directly into the large reservoirs downstream, there would be an energy cost in pumping it back up again.

The Riacho Verde Reservoir, one of the 4,000 small reservoirs in the region

GEORGE MAMEDEThe Riacho Verde Reservoir, one of the 4,000 small reservoirs in the regionGEORGE MAMEDE

The group’s studies also suggest that small reservoirs retain a good part of the sediment brought down by the water; if they did not it would accumulate in the large reservoirs, reducing their storage capacity. There is also evidence that the network of small reservoirs acts like a type of filter for the large reservoirs, retaining the pollution generated, above all by livestock farming.

Even with these benefits, however, the researchers warn that the construction of more small reservoirs – which is continuing in the region, albeit at a slower pace than in the past – needs to stop. A study led by Vanda Malveira, from UFC, and published in January in the Journal of Hydrologic Engineering, compared the growth in the network of reservoirs on the Upper Jaguaribe between 1961 and 2005 with computer-simulated networks. The researchers concluded that a network reaches an optimal use of its water resources when the sum of the capacity of the reservoirs reaches three times the volume of water that runs through the basin annually. Beyond this point, which was exceeded in the Upper Jaguaribe in the 1990s, there are no gains to be had by building new reservoirs. The water that would be stored by a reservoir downstream is simply transferred to another river upstream.

Intrigued with the similarities between the real reservoir network and the optimized virtual network, Mamede and Araújo sought the help of physicists who specialize in complex systems, Nuno Araújo, Christian Schneider and Hans Herrmann, from the Swiss Federal Institute of Technology in Zurich (ETH), to analyze the dynamics of water transportation in the basin. The team’s first challenge was to determine the location and maximum area of each reservoir in the basin, which facts were unknown for more than 95% of them. They logged the 3,798 reservoirs by satellite images provided by the National Space Research institute, which had been taken in exceptionally wet years (2004, 2008 and 2009). Then, using high resolution relief images taken by NASA’s Shuttle Radar Topography Mission, they reconstructed the course of each stream and river in the basin on the computer thereby discovering how each reservoir linked to the other.

To everyone’s surprise, they found that there was no typical average figure for the size of the reservoirs, whose area varies between 10,000 and 10 million square meters. The same goes for the number of their connections: there are isolated reservoirs, connected only with the Orós, as well as others linked to almost 400 reservoirs. “There is a very large heterogeneity; the system does not have a characteristic size,” explains Nuno Araújo. “We call it a scale-free network.”

061_Fisica_197With the complete map of the network, rainfall data from 131 meteorological stations spread throughout the region, and evaporation data from the Campos Sales station, the researchers created a hydrological model that computed how much water flowed into and out of each reservoir between 1991 and 2010. They discovered that on days of intense rainfall, because the reservoirs received water from one or more rivers and streams, but only had one outlet, they then overflowed one after another. In a cascade effect, the spillage from one reservoir led to others overflowing downstream. This cascade effect generally involved just two reservoirs, but could, with considerable frequency, affect 10, 100 or as many as 1,000 reservoirs.

The frequency with which cascades of different intensities happened follows a probability distribution that is typical of other phenomena, like earthquakes and blackouts in electricity networks, which physicists explain by means of models known as critically self-organized systems. They are so called because they are systems made from many parts interacting in an apparently random way, but from which emerge simple statistical laws, dictating that small changes have a possibility of causing great chain reactions in the system.

The group from the UFC is currently using this model to quantify the role of small reservoirs in lessening flood impact in large reservoirs. They are hoping to add more details to the model shortly, like the transportation of sediment and pollution and the structural integrity of the reservoirs, so that they can be used for assessing risk areas.

Scientific articles
MALVEIRA, V. T. C. et al. Hydrological impact of a high-density reservoir network in the semiarid north-eastern Brazil. Journal of Hydrologic Engineering. v. 17, p. 109-17. 2012.
MAMEDE, G.L. et al. Overspill avalanching in a dense reservoir network. Proceedings of the National Academy of Sciences. v. 109, p. 7.191-95. 2012.

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