1st International WDSA / CCWI 2018 Joint Conference, Kingston, Ontario, Canada – July 23-25, 2018

Comparison of One-Dimensional Unsteady Flow Models for

Simulating Pipeline and Pipe Network Hydraulics
Johnathan D. Nault, Ph.D.1
1Hydraulic Specialist, HydraTek & Associates, 216 Chrislea Road, Suite 204, Vaughan ON,
Canada, L4L 8S5, j.nault@hydratek.com

ABSTRACT
Unsteady (transient) flow models are frequently used for the analysis of water supply networks. Of
these, efficient quasi-steady models are well suited to long-term analyses, whereas the more
physically detailed rigid water column (RWC) and water hammer models better capture inertial
and compressibility effects. Though the models’ underlying physical assumptions are clear, their
ranges of validity seldom are. Improper model application can yield results that appear correct yet
are physically invalid, potentially misinforming decisions and increasing risk. To better understand
their ranges of applicability, this paper contrasts quasi-steady, RWC, and water hammer models on
their underlying formulations; moreover, indices are used to characterize physical error. Example
pipe systems compare the models’ for different events to demonstrate their performance. In addition
to exemplifying the need for assessing the validity of physical assumptions, unsteady flow metrics
are shown to provide utility in gauging model validity and guiding model selection.
Keywords: Unsteady flow, water hammer, pipe networks

1 Introduction and Background

Models are an invaluable tool for simulating the unsteady or transient hydraulics of water
distribution systems. Such systems often comprise hundreds to thousands of pipes with complex
interconnections, and models are essential to facilitating analyses [1]. Multiple types of transient
flow models are used, each with different physical accuracy and computational efficiency
(Figure 1). Quasi-steady (non-inertial unsteady) flow formulations (e.g., [2, 3]) neglect inertial and
compressibility effects (collectively referred to as dynamic effects), so they are best used for long-
term analyses – evaluating capacity, optimizing operations, or analyzing planning level decisions.
Comparatively, rigid water column (RWC; [4]) and water hammer models [5] better capture
dynamic effects resulting from boundary condition changes; they are suited to assessing structural
integrity. Though more physically detailed, water hammer and RWC models are also
computationally demanding. Practical analyses require models that are both accurate and efficient.
Conventionally, model selection is made according to the type of event simulated or the
objective of an analysis [6, 7]. Often though, such decisions are made without regard for physical
accuracy nor whether there are more economical approaches – for instance, efficient quasi-steady
models are commonly used to perform extended period simulations, yet the assumption of
negligible dynamic effects is clearly invalid when simulating boundary changes such as pump starts
and valve closures. Equally, water hammer models are used if dynamic effects are potentially
important. Despite recent computational advances, water hammer models still require significantly
more computational effort than RWC models, even for moderately-sized systems. Indeed, practical
analyses require models that are both physically accurate and computationally efficient. Key to
balancing this tension is understanding model validity.
 1st International WDSA / CCWI 2018 Joint Conference, Kingston, Ontario, Canada – July 23-25, 2018

Figure 1: Accuracy-efficiency trade-off for unsteady flow models [8]

Many authors have sought to characterize model validity by studying the importance of dynamic
effects. From the early work of [9] and [10] to later investigations (e.g., [6-8, 11, 12]), many have
explored the subject. However, characterizing dynamic effects remains difficult. Not only do
unsteady flows feature a range of complex phenomena (e.g., fluid-column separation [13]),
pressurized pipe systems also vary widely in their function, properties, and structure; nonetheless,
understanding the importance of dynamic effects is essential to model selection and thus
comprehensive analyses [1]. It is here that indicators find utility. To this end, the current paper
contrasts quasi-steady, RWC, and water hammer models on their physical modeling basis to
emphasize their roles in simulating unsteady pipe system hydraulics. Ultimately, practical analyses
require both accurate and efficient models, central to which is understanding the models’ physical
assumptions and ranges of validity.

2 Physical Modeling Basis

To illustrate how the models’ mathematical formulations relate to their assumptions, the
governing equations are reduced according to the models’ physical bases. Similar presentations can
be found elsewhere (e.g., [6, 11]), but the discussion here reemphasize the importance of the
underlying physical assumptions. One-dimensional unsteady pressurized flow is described by a pair
of partial differential equations [14]:
a 2 ∂Q ∂H ∂H
0= + +V − V sin(α ) (1)
gA ∂x ∂t ∂x
∂H 1 ∂Q 1 ∂Q f
0= +J+ + V , J= |Q |Q (2)
∂x gA ∂t gA ∂x 2 gDA2
Equation (1) is the momentum equation, and Equation (2) is the continuity equation, where
Q = flow (m3/s), V = velocity (m/s), H = head (m), a = wave speed (m/s), g = 9.81 m/s2, A = conduit
area (m2), α = pipe inclination, J = unit head loss, f = friction factor, D = pipe diameter (m),
x = distance along the conduit (m), and t = time (s). Equation (1) represents the effects of fluid
compressibility and conduit elasticity, and Equation (2) relates energy losses and inertial effects to
spatial changes in head. Though often omitted, Equations (1) and (2) include the convective
acceleration terms V•∂H/∂x and V•∂Q/∂x.
Water hammer models capture compressibility, inertial, and viscous resistance (friction) effects.
Their only assumption of Equations (1) and (2) is that V•∂H/∂x and V•∂Q/∂x are relatively small
compared to the other terms, and dropping these simplifies the expressions to
 1st International WDSA / CCWI 2018 Joint Conference, Kingston, Ontario, Canada – July 23-25, 2018

a 2 ∂Q ∂H ∂H 1 ∂Q
0= + , 0= +J+ (3)
gA ∂x ∂t ∂x gA ∂t
In addition to the convective terms, RWC models additionally assume that changes at the
boundaries of a system occur slowly enough such that the fluid is approximately incompressible
(that is, ∂Q/∂x ≈ 0), leaving only the momentum expression that further reduces to
L dQ n −1
0 = H 2 − H1 + F (Q ) + , F (Q ) = K Q Q (4)
gA dt
where F(Q) = head loss (m), L = pipe length (m), K = resistance coefficient (sn/m3n-1), and
n = exponent that depends on the head loss model (n = 2 for the Darcy-Weisbach model and
n = 1.852 for the Hazen-Williams model). Lastly, quasi-steady and steady-state models further
presume that ∂Q/∂t is small enough that even inertial effects are negligible (that is, ∂Q/∂t ≈ 0). For
∂Q/∂t = 0, Equation (4) becomes
0 = H 2 − H1 + F (Q ) (5)
Central to the above simplifications are assumptions of the underlying physics, namely,
compressibility effects and inertial effects. Indeed, the models are well defined by their treatment of
the governing equations, but their ranges of validity are not always so clear. Two questions arise in
this regard: how important are dynamic effects, and which type of model captures the appropriately
physical features? In answering these, unsteady flow characterization is needed to classify the
transient flow regimes and thus guide model selection.

3 Unsteady Flow Characterization

Parallel to the types of models are the unsteady flow regimes [6]. Unsteady-compressible flow is
characterized by high frequency fluctuations, where compressibility and inertial effects are the
predominant mechanisms. Comparatively, compressibility effects are less important (albeit still
present) under unsteady-incompressible and quasi-steady flows. Even inertial effects are negligible
for the latter, as head losses dominate the hydraulics. Unlike the different types of models, the
unsteady flow regimes are not separated by discrete boundaries; rather, there are transitional regions
where compressibility, inertial, and even unsteady friction effects become progressively more
important (the latter is not considered here; Figure 2). Understanding the transitional regions is
essential to model selection.

Figure 2: Relative importance of dynamic effects [8]

 1st International WDSA / CCWI 2018 Joint Conference, Kingston, Ontario, Canada – July 23-25, 2018

A number of indicators have been developed for characterizing unsteadiness and thus the
transitional regions (Table 1). Because the degree of unsteadiness depends on the boundary change
magnitude (∆Q or ∆H) and duration (TBC), unsteady flow metrics tend to describe one or both of
these. The Joukowsky head HJ, Allievi parameter ρ, and attenuation index R consider the magnitude
of an event, whereas the characteristic time TC and Allievi parameter θ relate to the time scale of a
system. Both types of indicators are complementary, as well as those that encompass the magnitude
and duration of an event (HA, ϕC, TW, Γ, ϕA, and ϕR). For further details on each metric, the reader is
referred to the references in Table 1 along with [7, 8, 11, 12]. Certainly unsteady flow indicators are
helpful, but it nonetheless remains difficult to define firm boundaries beyond which one model
should be applied over another. This is largely because pipe systems vary in structure and
properties; moreover, there are a range of possible types of boundary changes. The following
section presents two examples that contrast the models and illustrate the utility of the indicators.

Table 1: Metrics for characterizing unsteady pipe flow

Metric Reference Expression Magnitude Duration
Characteristic Period - TC = 2 L / a •
L ∂Q L ∆Q
Acceleration Head - HA = ≈ • •
gA ∂t gA TBC
a
Joukowsky Head [9] HJ = ∆Q •
gA
aV0
Compressibility Index [10] ρ= •
2 gH 0
Time Period Index [10] θ = TBC / TC •
max ∆U
Compressibility Index* [6] φC = • •
max ∆T
F (Q ) fL Q
Attenuation Index [15] R=
HJ
=
2 DAa
•
LQ0
Water Column Acceleration Period [11] TW = •
gAH 0
θ gAH 0TBC
Inertial Index [11] Γ= = • •
ρ LQ0
L ∂H L ∂Q
Absolute Dynamic Effects Indicator [8] φA = + • •
a ∂t gA ∂t
φA
Relative Dynamic Effects Indicator [8] φR = • •
F (Q ) + φ A
*U = internal energy, T = total kinetic energy, and the subscript 0 indicates initial conditions.

4 Simulation of Unsteady Flow

Model selection is often clear for analyses concerning long-term capacity or short-term structural
integrity. Rarely though is this the case for intermediate conditions, particularly unsteady-nearly
incompressible flow. Below, two examples contrast the different types of models for such
conditions, and unsteady flow indicates are used to characterize the system responses. It is
recognized that quasi-steady models are ill-suited to simulating boundary changes; nonetheless,
they are included below to illustrate the dynamic effects that are otherwise neglected.
 1st International WDSA / CCWI 2018 Joint Conference, Kingston, Ontario, Canada – July 23-25, 2018

4.1 Example 1 – Water Transmission Main

In addition to hydraulic capacity and structural integrity, unsteady flow simulation is essential to
analyzing operational changes. Routine operations are more frequent than power failure and rapid
valve closure events, and though less extreme, they can still generate excessive pressures and cyclic
loading conditions. Consider a controlled pump stop for a water transmission pipeline (Figure 3),
which is based on a real water supply pipeline. The operation involves closing the pump’s discharge
valve over TBC according a cubic closure curve, and then the pump is shut off. Here the objective is
to contrast the models’ validity.

PSV:
HGL = 140 m

Segment 1 Segment 2 Segment 3

R1: 100 m L = 6,000 m L = 12,400 m L = 33,600 m R2: 115 m
D = 1,050 mm D = 750 mm D = 600 mm
f = 0.015 f = 0.015 f = 0.015
Pumping Station:
ܳ௣௨௠௣ = 190 L/s, ‫ܪ‬௣௨௠௣ = 91.4 m
Figure 3: Example 1 – 52 km long water transmission pipeline
Figure 4 illustrates the system response for a range of TBC and a as simulated by the different
types of models. Compressibility and inertial effects are clearly important for small TBC, which is
reflected in the larger HA, Γ, ϕA, and ϕR values (ϕ is excluded, as it cannot be estimated a priori [6]).
A water hammer model is necessary to capture the resulting unsteady flow dynamics. For moderate
TBC, inertial effects and head losses dominate the response with lower values for HA, Γ, ϕA, and ϕR,
so an RWC model provides a reasonable representation. Ultimately, the models converge for large
TBC due to negligible dynamics effects. In order for a quasi-steady model to be physically valid
though, TBC must be greater than TC by several orders or magnitude; certainly this is impractical for
routine operations, thereby exemplifying the limited range of validity for quasi-steady models.
Further on the matter of model validity, Figure 5 compares the head extrema error for quasi-
steady and RWC models (relative to a water hammer model) against Γ and ϕA. Only these two
metrics are considered here, as both reflect the magnitude and duration of boundary changes. Not
surprisingly, the plots are comparable. Because Γ and ϕA both consider TBC and ∆Q in a similar
fashion, the two indicators are inversely proportional, with the only difference being that ϕA
additionally incorporates a. The absolute dynamic effects indicator also has clearer physical
meaning; it characterizes the terms omitted from quasi-steady and RWC models, making it suitable
for gauging model error. For instance, the quasi-steady model error is roughly 2 m for ϕA = 2 m, and
the RWC model error is slightly less than ϕA for ϕA ≤ 5 m. Altogether, these and other indicators,
while simplifications, provide insight into the degree of unsteadiness for a system.
4.2 Example 2 – Water Distribution System
Unlike pipelines, pipe networks feature arrays of interconnected pipes with varied structures,
pipe properties, and boundary conditions. Their responses are markedly more complex, making it
difficult to determine which type of model is the most suitable. Here the models are contrasted for a
sequential pump start-pump stop operation in a complex network, a water distribution system
(Figure 6; [16]); moreover, two scenarios are considered, one with no surge protection and another
with a hydropneumatic air chamber at the source pump station (a consistent a = 1,000 m/s is used
throughout the system for simplicity). The objective is to demonstrate the models’ validity with and
without dedicated surge protection.
 1st International WDSA / CCWI 2018 Joint Conference, Kingston, Ontario, Canada – July 23-25, 2018

200 200
Quasi-Steady Model (∆t = 1 s) Quasi-Steady Model (∆t = 1 s)
175 RWC Model (∆t = 1 s) 175 RWC Model (∆t = 1 s)
Pump Station Discharge Head (m)

Pump Station Discharge Head (m)

Water Hammer Model (∆t = 0.1 s) Water Hammer Model (∆t = 0.1 s)
150 150

125 125

100 100
Parameters: Parameters:
= 1,000 m/s = 100 s = 1,000 m/s = 100 s
75 75
Unsteady Flow Indicators: Unsteady Flow Indicators:
50 = 104 s = 1.2 50 = 208 s = 2.4
= 36.6 m = 27.5 m = 36.6 m = 13.8 m
= 0.42 = 0.96 = 0.21 = 0.48
25 = 112 s = 2.3 25 = 112 s = 2.3
= 55 m = 0.63 = 74 m = 0.70
0 0
0 500 1,000 1,500 2,000 0 500 1,000 1,500 2,000
Time (s) Time (s)

180 180
Parameters: Parameters:
= 1,000 m/s = 1,000 s = 1,000 m/s = 1,000 s
Pump Station Discharge Head (m)

Pump Station Discharge Head (m)

170 170
Unsteady Flow Indicators: Unsteady Flow Indicators:
= 104 s = 1.2 = 208 s = 2.4
= 3.7 m = 27.5 m = 3.7 m = 13.8 m
160 = 0.42 = 9.6 160 = 0.21 = 4.8
= 112 s = 23 = 112 s = 22.8
= 5.5 m = 0.15 = 7.4 m = 0.19
150 150

140 140

130 Quasi-Steady Model (∆t = 1 s) 130 Quasi-Steady Model (∆t = 1 s)

RWC Model (∆t = 1 s) RWC Model (∆t = 1 s)
Water Hammer Model (∆t = 0.1 s) Water Hammer Model (∆t = 0.1 s)
120 120
0 500 1,000 1,500 2,000 0 500 1,000 1,500 2,000
Time (s) Time (s)

180 180
Parameters: Parameters:
= 1,000 m/s = 10,000 s = 1,000 m/s = 10,000 s
Pump Station Discharge Head (m)

Pump Station Discharge Head (m)

170 170
Unsteady Flow Indicators: Unsteady Flow Indicators:
= 104 s = 1.2 = 208 s = 2.4
= 0.4 m = 27.5 m = 0.4 m = 13.8 m
160 = 0.42 = 96 160 = 0.21 = 48
= 112 s = 230 = 112 s = 228
= 0.55 m = 0.02 = 0.74 m = 0.02
150 150

140 140

130 Quasi-Steady Model (∆t = 1 s) 130 Quasi-Steady Model (∆t = 1 s)

RWC Model (∆t = 1 s) RWC Model (∆t = 1 s)
Water Hammer Model (∆t = 0.1 s) Water Hammer Model (∆t = 0.1 s)
120 120
0 2,000 4,000 6,000 8,000 10,000 12,000 0 2,000 4,000 6,000 8,000 10,000 12,000
Time (s) Time (s)

Figure 4: Example 1 – comparison of model performance for different TBC and a

Figure 5: Example 1 – head extrema error for quasi-steady and RWC models
 1st International WDSA / CCWI 2018 Joint Conference, Kingston, Ontario, Canada – July 23-25, 2018

Figure 6: Example 2 – water distribution system [16]

Time series results for the high lift pump station discharge head are shown in Figure 7. As with
Example 1, a quasi-steady model fails to adequately capture the resulting transient hydraulics due to
neglecting dynamic effects (again, quasi-steady models are not suitable here, but the comparison
highlights unsteady hydraulics that are otherwise neglected). Without the surge tank, an RWC
model predicts the head extrema with modest error. Interestingly though, the RWC and water
hammer models compare well for the head extrema if the source pump station is equipped with a
surge vessel; in effect, this is because surge tanks provide additional storage to accommodate
changes in the system, thereby moderating the degree of unsteadiness. It is quite likely that even
shorter operation durations can be considered while an RWC model remains reasonably valid. For
this reason, one must consider not only the system, but also key network elements that otherwise
alter the system response.
155 155
*No Surge Tank at Parameters: *Surge Tank at Source Parameters:
Source Pump Station = 1,000 m/s = 60 s Pump Station = 1,000 m/s = 60 s
150 150
Pump Station Discharge Head (m)

Pump Station Discharge Head (m)

Unsteady Flow Indicators: Unsteady Flow Indicators:

=6 s = 0.01 =6 s = 0.01
145 = 14.3 m = 62.8 m 145 = 14.3 m = 62.8 m
= 0.51 = 11 = 0.51 = 11
= 14 s = 21 = 14 s = 21
140 = 17.1m = 0.97 140 = 17.1m = 0.97

135 135

130 130

125 Quasi-Steady Model (∆t = 5 s) 125 Quasi-Steady Model (∆t = 5 s)

RWC Model (∆t = 1 s) RWC Model (∆t = 1 s)
Water Hammer Model (∆t = 0.02 s) Water Hammer Model (∆t = 0.02 s)
120 120
0 100 200 300 400 500 600 0 100 200 300 400 500 600
Time (s) Time (s)

Figure 7: Example 2 – WTP HLPS discharge head time series for sequential pump start and pump
stop operations
 1st International WDSA / CCWI 2018 Joint Conference, Kingston, Ontario, Canada – July 23-25, 2018

4.3 Implications of Model Selection

Of the three formulations, water hammer models clearly provide the greatest physical accuracy.
By comparison, quasi-steady models simply cannot capture the dynamics of even moderately
unsteady flows. Quite naturally, one question arises – why not simply always use a more physically
accurate model? Clearly this is not practical. Greater model accuracy, be it numerical or physical,
comes at the expense of greater computational effort; to economize modeling, one must resolve the
accuracy-efficiency tension. Crucial to this is understanding the importance of dynamic effects,
which can be accomplished using the unsteady flow indicators. Of course, the indicators themselves
are approximations with inherent differences and limitations.
To describe unsteadiness, the indicators characterize the magnitude and duration of boundary
changes. The key distinction between different metrics is how they consider one or both of these
facets, a matter that directly relates to their use (it is also worth noting that none of the indicators
here consider the benefits of surge protection nor unsteady flow phenomena; e.g., pressure wave
superposition, nor fluid column separation). For instance, HJ represents the head change following
an abrupt flow change ∆Q, thereby describing the potential surge, but the metric only applies to
boundary changes that occur on a time scale order of TC. Such indicators are best paired with others
that are complementary (e.g., ρ and θ). Alternatively, one can employ metrics that consider both the
magnitude and duration of a boundary change, such as HA or Γ. These are inversely proportional (as
demonstrated in Example 1); moreover, the former has clearer physical meaning, and it is
generalized by ϕA and complemented by ϕR. Together, ϕA and ϕR relate to terms in the governing
equations that are omitted by each model, making them ideal for gauging model validity.
Though ϕA and ϕR are helpful for describing model validity, there remains the question of when
to use one model over another. With a means of gauging dynamic effects, model selection should
ideally be made according to an acceptable level of error [4, 8]. For example, ϕA ≈ 1 m indicates
that quasi-steady models may have a pipe head difference error of approximately 1 m, yet an RWC
model is likely suitable. If, however, ϕA and ϕR are larger (that is, dynamic effects are both present
and relatively important), then a water hammer model may even be necessary. Whether this is
important depends on the objective of an analysis. Altogether, balancing the accuracy-efficiency
tension requires an understanding of the importance of dynamic effects, which can be accomplished
using unsteady flow indices.

5 Concluding Remarks
Model selection is fundamental to simulating transient flow in distribution systems. There are
multiple types of models, each with different accuracy and computational features. Quasi-steady
models are efficient, yet they simply cannot provide a representative solution of even moderately
unsteady flows. Conversely, RWC and water hammer models have greater physical accuracy but
also greater computational cost. Balancing the accuracy-efficiency tension is central to practical
simulation; the relevant physics should be captured while minimizing computational effort.
Key to appropriate model selection is understanding how the models’ ranges of validity relate to
the objective of an analysis. In this paper, the models’ physical bases were first contrasted, and
unsteady flow indicators were reviewed for characterizing the importance of dynamic effects. Those
indicators that consider the magnitude and duration of boundary changes are more useful, which
was illustrated via two examples. Moreover, model validity also depends on the particular system
properties. Further work is still needed to refine the unsteady flow metrics, ideally allowing them to
be more widely used. Another approach to resolving the model selection task is adaptive hybrid
modeling (see Nault and Karney, 2016; Nault, 2017). Ultimately, one must economize analyses, but
equally important is understanding the implications of neglecting dynamic effects to avoid
overlooking potentially critical system hydraulics.
 1st International WDSA / CCWI 2018 Joint Conference, Kingston, Ontario, Canada – July 23-25, 2018

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