Fractal Geometry in Designing and Operating Water Networks

Abstract

Fractals are self-similar sets that cannot be easily described by classical geometry. Fractal sets have been implemented in almost all areas of human activity since they were introduced to science by Mandelbrot in 1982. For the last 10 years, the interest in fractal geometry has increased by the issues connected with water distribution networks (WDNs). The aim of this paper was to review the application of fractal geometry in designing and operating WDNs. Treating a WDN as a fractal pattern enables its description and classification, simplifies the assessment of a network reliability, helps to solve the problems of routing and dimensioning WDN, as well as enables to select the places to locate measurement points in a network to control water quality, pressure in pipes and water flow rate. Moreover, the application of tree-shaped fractal patterns to reflect WDNs helps to solve the problems of their optimization. Fractal geometry can be also applied to investigate the results of WDNs failures connected with leakage of water to the ground. Using fractal dimension of a pattern created by points reflecting places of water outflow on the soil surface after a prospective pipe breakage enables to determine the zone near a pipe, where the outflow of water on the soil surface is possible. It is an important approach for the security of humans and existing infrastructure. Usage of fractal geometry in description, optimisation and operation analysis of WDNs still continues, which confirms the efficiency of fractal geometry as a research tool. On the other hand, it can be supposed that fractal geometry possibilities have still not been fully used.

Fractal Characteristics of Water Outflows on the Soil Surface after a Pipe Failure

“…Fractal geometry has become a very helpful tool used both for the characterization of complex microstructures of porous media and in theoretical analyses to determine the principles of liquid flow through these media. In recent years, the scope of interest in fractal geometry has expanded to include issues related to the design and operation of water supply networks [58]. These are not the only examples of the use of fractal geom-etry as a research tool.…”

A Community-Structure-Based Method for Estimating the Fractal Dimension, and its Application to Water Networks for the Assessment of Vulnerability to Disasters

“…Diao et al (2017) developed a fractal-based analysis to identify critical pipes in WDSs. A review of the topic can be found in Iwanek et al (2020), while recently, Caldarola and Maiolo (2020) investigated the opportunities of using fractal theory for the description and interpretation of data and dynamics of real complex systems. Finally, a review of the topic of fractal dimension of complex networks has been done by Wen and Cheong (2021).…”

Fractal Analysis of the Optimal Hydraulic Gradient Surface in Water Distribution Networks

“…Furthermore, fractal behavior is not restricted to rough surfaces. It has been shown that a WDN may show fractal properties in its topological structure that may be applied in the analysis of the resilience, reliability, water quality, and operational conditions, as well as in classification, risk assessment, and sectorization procedures (Diao et al 2017;Di Nardo et al 2017;Kowalski et al 2019;Iwanek et al 2020;Kowalska et al 2020;Giudicianni et al 2021). In this context, WDNs have been examined employing graph theory concepts by modeling their underlying topological structure as directed or undirected graphs whose vertices must be covered with the smallest number of boxes or clusters of a certain size (Diao et al 2017;.…”