The calculation of hydraulic state variables for a network is an important task in managing the distribution of potable water. Over the years the mathematical modeling process has been improved by numerous researchers for utilization in new computer applications and the more realistic modeling of water distribution networks. But, in spite of these continuous advances, there are still a number of physical phenomena that may not be tackled correctly by current models. This paper will take a closer look at the two modeling paradigms given by demand- and pressure-driven modeling. The basic equations are introduced and parallels are drawn with the optimization formulations from electrical engineering. These formulations guarantee the existence and uniqueness of the solution. One of the central questions of the French and German research project ResiWater is the investigation of the network resilience in the case of extreme events or disasters. Under such extraordinary conditions where models are pushed beyond their limits, we talk about deficient network models. Examples of deficient networks are given by highly regulated flow, leakage or pipe bursts and cases where pressure falls below the vapor pressure of water. These examples will be presented and analyzed on the solvability and physical correctness of the solution with respect to demand- and pressure-driven models.

Calculating the flow in hydraulic networks has a long history starting with
the work presented by

In hydraulic modeling the simplified topological structure of a water
distribution network is described by a directed graph. This graph represents
pipe sections as links and pipe junctions as nodes. The mathematical
description of this graph is given by the incidence matrix

Water distribution networks have a looped structure and the system state is
described by the potential at the nodes (head) and the current for the links
(flow). The system equations are given by the following sets of equations:
first the mass balance at every node

Here the incidence matrix is divided into two parts.

The inverse relation is defined as

The constant

Based on the works of

For linear systems Maxwell's theorem states that the distribution of current
or flow which gives a minimum value to the function

For linear systems

However, in recent years new publications have shown that the loop method
gives an efficient alternative to the hybrid approaches.

The demand-driven modeling approach has considerable shortcomings under complex boundary conditions and is not able to realistically model mechanisms that are driven by pressure differences. Two of the most important phenomena are pressure-dependent demands at consumption nodes and pressure-dependent leakage for pipe ruptures.

In the case of pressure-dependent demand, experience has shown that under
certain conditions the demand-driven model can lead to non-physical
solutions. This is the case in pressure-deficient networks where, under
realistic conditions, the demand cannot be met at certain consumer nodes.
From hydrostatics, it is known that
the maximum flow volume depends on the difference between the nodal and
atmospheric pressure. To take this into account the pressure-driven modeling
approach relaxes the demand boundary conditions and the fixed consumption is
replaced by the set of inequality conditions

By far the most popular approach to handle the degree of freedom introduced
by the pressure-dependent formulation is the introduction of an emitter
function

One of the first publications on the topic by

In the case of pressure-dependent discharges for pipe ruptures,

Using a general emitter function, a modified set of equations has been
published by

As for the demand-driven model, it is also possible to formulate the content
and co-content problems for the pressure-driven approach. Following

The pressure-dependent Co-Content Model is given as

From the literature the notion of deficient networks can take a number of
different definitions. These definitions may be divided into model,
mathematical and physical deficiencies. Model deficiencies are errors in the
creation, conversion or transfer of the network graph. A mathematical
deficiency can be defined as a maximally connected network where, due to some
boundary condition, the set of feasible solutions is reduced to the empty set
or the solution is not unique. In contrast to mathematical deficiencies, in
the case of a hydraulic deficiency a unique solution exists, but it is
physically incorrect. With respect to the two modeling paradigms presented in
Sects.

In conclusion, this paper has given a summary of the current state of water distribution network modeling, looking into the classical approaches using mass balance and energy equations, as well as optimization approaches that allow one to make assumptions about the properties of the solution space. These formulations have been given for both the framework of demand-driven modeling with very strict constraints and pressure-driven modeling where the constraints have been relaxed to give more realistic results.

Looking at the two modeling paradigms, four phenomena of deficient networks of interest to the ResiWater Project have been analyzed with respect to solvability and physical accuracy. It can be concluded that, although the pressure-driven approach is far superior to demand-driven modeling in cases like pressure-controlled demands and leakage, there still exist model deficiencies in cases where pressure drops below a physically realistic level.

There are no data available since the article does not refer to any data sets.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Computing and Control for the Water Industry, CCWI 2016”. It is a result of the 14th International CCWI Conference, Amsterdam, the Netherlands, 7–9 November 2016.

The work presented in the paper is part of French–German collaborative research project ResiWater that is funded by the French National Research Agency (ANR; project: ANR-14-PICS-0003) and the German Federal Ministry of Education and Research (BMBF; project: BMBF-13N13690). Edited by: Edo Abraham Reviewed by: three anonymous referees