Organization and scaling in water supply networks

Likwan Cheng and Bryan W. Karney

Phys. Rev. E 96, 062317 – Published 29 December 2017

Abstract

Public water supply is one of the society's most vital resources and most costly infrastructures. Traditional concepts of these networks capture their engineering identity as isolated, deterministic hydraulic units, but overlook their physics identity as related entities in a probabilistic, geographic ensemble, characterized by size organization and property scaling. Although discoveries of allometric scaling in natural supply networks (organisms and rivers) raised the prospect for similar findings in anthropogenic supplies, so far such a finding has not been reported in public water or related civic resource supplies. Examining an empirical ensemble of large number and wide size range, we show that water supply networks possess self-organized size abundance and theory-explained allometric scaling in spatial, infrastructural, and resource- and emission-flow properties. These discoveries establish scaling physics for water supply networks and may lead to novel applications in resource- and jurisdiction-scale water governance.

Received 14 June 2017

DOI:https://doi.org/10.1103/PhysRevE.96.062317

Physics Subject Headings (PhySH)

Scaling laws of complex systems Transport in networks Transportation networks

Interdisciplinary PhysicsNetworks

Authors & Affiliations

Likwan Cheng^1,* and Bryan W. Karney²

¹Physical Science Department, City Colleges of Chicago, Chicago, Illinois 60601, USA
²Department of Civil Engineering, University of Toronto, Toronto, Ontario M5S 1A4, Canada

^*lcheng6@ccc.edu

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Issue

Vol. 96, Iss. 6 — December 2017

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Images

Figure 1
Internetwork organization and intranetwork structure of public water supplies. (a) Geographic ensemble of PWSs in the U.S. state of Wisconsin (circles, $n = 582$ ), with consumption $M$ shown in annually averaged daily volume of million gallons per day (mgd). (b) Sample 3D consumer density profile and (c) 2D PWS distribution network at the city of Ashland [21]. (d, A) Complementary cumulative distribution function (CCDF) of $M$ , with lognormal (LN) distribution fit (Kolmogorov-Smirnov test, $p = 0.54$ ) and asymptotic power-law (PL) fit $P \sim M^{- α + 1}$ , the latter with best-fit $α = 2.3$ (KS test, $p = 0.13$ ); (d, B) probability density function (PDF) of log-transformed $M$ with normal distribution fit. (e) Rank-size distribution of $M$ compared to Zipf's inverse-rank law $M (r) \propto 1 / r$ , where $r$ is rank, and fitted with the discrete generalized beta distribution (DGBD) $M (r) \propto {(n + 1 - r)}^{β} / r^{α}$ [22], with best-fit $(α, β) = (0.98, 0.94)$ . (f) CCDF of node degrees $k$ of the network at Ashland (blue circles) with exponential fit $P = exp (- a k)$ , with best-fit $a = 0.4$ ; node-degree distributions (black squares) for four progressively less ordered PWS networks from Ref. [23].
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Figure 2
Scaling in public water supply networks. Log-log plots showing empirical data (colored dots; each is a PWS), best-fits (solid line), ideal predictions (short dashed-line), and isometric scaling (long dashed-line); $b$ is the scaling exponent of $M$ . In (c), $d_{\mod}$ and $d_{\max}$ are the mode and maximum main diameters, respectively. In (a) and (c), multiple data sets are vertically shifted for clarity of display. Axis values indicate 10-based exponentials.
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Figure 3
Geometric relations among the transmission main, distribution main, lateral service line, consumer, and service area.
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Figure 4
(a) Comparison of the fits to the data with the nonlinear model [Eq. (B3)] and the linear approximation model [Eq. (B6)] for the scaling of network energy $E$ . (b) (top to bottom) The standardized data and the standardized residues from the nonlinear fit and the linear approximation fit, each shown with a (log-transformed) normal distribution fit. Approximate normalcy is observed in both the data and the residue distributions.
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Figure 5
The fractional factor $Φ$ in network head loss $h_{n}$ in Eq. (C3) for a typical network branching ratio $β = 16$ (blue); the scaling exponent $b_{Φ}$ in the relation $Φ = M^{b_{Φ}}$ for a typical maximum number of network hierarchical levels $t = 10$ (red), showing the convergence of $b_{Φ}$ to 0.
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Figure 6
Comparison of scaling exponents for energy dissipations under Hazen-Williams and Ohm's mechanisms. The Hazen-Williams exponent curve is based on Eq. (D2) and converges at $b = 0.823$ ; the Ohm's exponent curve is based on Eq. (D4) (derived from Eq. (3) in Ref. [14]) and converges at $b = \frac{7}{6}$ . Both curves are calculated for a network with the maximum number of hierarchical levels of $t = 10$ .
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