Maths for understanding water flow

Some places have higher levels of precipitation while others have higher levels of dryness, so it is no surprise that maps showing climatic variations will vary greatly. The frequency of rainfall and its partial evaporation, the nature of soils and the way they are used and water flow into rivers are all parameters that are impacted. These parameters initiate a cause and effect chain of events that influence the capacity of our rivers, industrial production, and agriculture and could extend to the management of water, flood-risk areas and periods of dryness. Having the capability to predict these evolutions has become a major challenge for hydrology. Each river basin has individual characteristics that have a unique effect on the water cycle. A colossal amount of data is required in order to correctly calibrate ecosystem models. However, there may be another way of predicting the behaviour of river basins across the world while at the same time eliminating the need for vast and tedious amounts of field data. This “universal response” can be found in a simple equation, the maximum power principle. Intuition is bold but nature tends to respond favourably to empirical methods. Put another way, up to now there is nothing to prove that this equation doesn’t work.  

There are three main possible outcomes for rainwater once it reaches a river basin. It can evaporate, it can flow into rivers or seep into the ground. The extent of these three outcomes depends on factors such as temperature, the frequency of precipitation and the properties of basins (type of ground cover, presence of vegetation, etc.). All of these factors are naturally influenced by weather variations. Global warming will naturally exacerbate trends that are already unbalanced. Rainfall will become more intense in regions where it is already high and will become rarer in areas that are already endlessly subjected to periods of drought. These developments will have important hydrological and environmental consequences, mainly due to modifications in river flow volumes. Higher, more extreme flow volumes will increase the risk of flooding. Conversely, lower flow volumes will limit access to drinking water, agriculture, industrial exploitation and navigation. “The urgent question is now to succeed in determining the impact of climate change on river basins and river flow volumes”, states Martijn Westhoff, a post-doctoral student at the HECE unit (Hydraulics in Environmental and Civil Engineering) in the Faculty of Applied Sciences of the University of Liege and the leading author of an article published in Hydrology and Earth System Sciences (HESS) (1). “In order to do this, we usually create models that are calibrated by means of surveyed data from the field. We can then modify certain variables and simulate changes to water flow. But these models have significant limitations. Most of them only take into account variations in precipitation and temperature. They do not take account of the evolution of river basin properties”. However, changes in soil structure but also their cover (presence of dense vegetation, etc.) will have a significant impact on evaporation and seepage and therefore also on water flow. “The main problem with these complex models involves their calibration, explains Benjamin Dewals, a lecturer in hydraulic engineering at ULg and co-author of the publication. “In order to accurately reflect reality, they must be extremely detailed. In order to achieve this, an enormous amount of data over several years must be gathered in order to be integrated into the models and only then can simulations even begin. And in the case of many river basins, which are often barely accessible, we have very few measurements. At the same time, there is a real need to predict their evolution for the coming decades. And hydrological models are the only tools that make it possible to assess the risk of flooding or drought, or to improve water quality management based on water flow levels”.

Curves rivers

From the particular to the universal

Certainly, the creation of a complex and detailed model, capable of reproducing all the processes while taking into account the countless parameters such as the type of vegetation, erosion or the chemical nature of soils is an attractive prospect. But the approach is a laborious one and Martijn Westhoff chose a second, complementary option, which, though not as precise, seems to be a more realistic method at the current time. “The idea is to search for a model that will provide a greater understanding and more global answers”, says the engineer. “It is no longer a question of studying the particularities of every river basin and their individual responses, or the evolution of each plant in relation to an increase in temperature, but rather to identify an overall ‘universal’ principle to which nature responds, a principle that each basin will follow in order to adapt to later changes”.

(1) Westhoff, M., Zehe, E., Archambeau, P., and Dewals, B.: Does the Budyko curve reflect a maximum-power state of hydrological systems? A backward analysis, in Hydrology and Earth System Sciences, 20, 479-486, doi:10.5194/hess-20-479-2016, 2016.

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