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ADE | Quasi-static and dynamic fracture toughness tests
2 ∆𝑎 𝐾 𝐾′√(∆𝑎−𝑥)
𝐺 = lim ∫ 𝐼 𝐼 𝑑𝑥 (4.17)
𝐼 ∆𝑎→0(1+𝑣)𝜇∆𝑎 √2𝜋𝑥 √2𝜋
0
∆𝑎 can be very small, such that ∆𝐾′ can be made small enough in comparison to 𝐾 , and as a
𝐼
result of which ∆𝐾′ can be neglected.
𝐾2 ∆𝑎√∆𝑎−𝑥
𝐺 = lim 𝐼 ∫ 𝑑𝑥 (4.18)
𝐼 ∆𝑎→0(1+𝑣)𝜇∆𝑎 𝑥
0
In order to solve the integral, putting 𝑥 = ∆𝑎𝑠𝑖𝑛2𝛼. When 𝑥 = 0 and 𝛼 = 0 and when 𝑥 =
𝜋
∆𝑎,𝛼 = 𝑎𝑛𝑑 𝑑𝑥 = ∆𝑎 2𝑠𝑖𝑛𝛼 𝑐𝑜𝑠𝛼 𝑑𝛼
2
𝐾2 𝜋/2 (∆𝑎−∆𝑎𝑠𝑖𝑛2𝑎)
𝐺 = lim 𝐼 ∫ √ ∆𝑎2sin𝑎𝑐𝑜𝑠𝑎𝑑𝑎
𝐼 ∆𝑎→0(1+𝑣)𝜋𝜇∆𝑎 ∆𝑎𝑠𝑖𝑛2𝑎
0
𝐾2∆𝑎 𝜋/2
𝐺 = 𝐼 ∫ 2𝑐𝑜𝑠2𝑎𝑑𝑎
𝐼 (1+𝑣)𝜋𝜇∆𝑎
0
(4.19)
𝐾2 𝜋/2
𝐺 = 𝐼 ∫ 2𝑐𝑜𝑠2𝑎𝑑𝑎
𝐼 (1+𝑣)𝜋𝜇
0
𝐾2 𝜋
𝐼
𝐺 =
𝐼 (1+𝑣)𝜋𝜇 2
𝐾22(1+𝑣) 𝜋
𝐼 (4.20)
𝐺 =
𝐼 (1+𝑣)𝜋𝐸 2
𝐾2
𝐼 (4.21)
𝐺 =
𝐼 𝐸
4.3.3 - Quasi-static mode I fracture toughness test results
The load-displacement curves of granite which represent the rock characteristics were directly
obtained from SCB fracture toughness tests. Figure 4.10 shows the typical load-displacement
curves of Australian granite with different loading rates at various temperatures obtained in
this study. After the elastic stage, the rock suddenly broke in a typical brittle failure. Each load-
displacement curve exhibits a slowly increasing portion until a peak followed by a dramatically
falling post-failure portion indicating a brittle fracture. The turning point at the peak force in
Figure 4.10 denotes the stable-unstable fracture transition of the specimen. Figure 4.11
illustrates typical failed specimens for each temperature group and fracture surface of a
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Table 4.4 summarises the failure loads and the corresponding fracture toughness values for all
sets of the specimens at different temperatures and loading rates. More detailed results of the
fracture toughness are depicted in Figure 4.12a. The relation between the mode I quasi-static
fracture toughness, loading rate and temperature for CCNSCB specimens treated at various
temperatures and under different loading rates is depicted in Figure 4.12b. It can be seen from
Figure 4.12 that the quasi-static mode I fracture toughness and energy-release rate (given by
Equation 4.21) at the same heat-treatment temperature increased linearly with the loading rate.
As in the case with increasing loading rate, the load required to fail the specimen increased
which resulted in a rising trend of the fracture toughness of the rock as they are dependent on
each other. The cracks which were mostly formed by intergranular fractures under low loading
rates caused rougher fracture surfaces, when compared to that of the samples failed under high
loading rates. However, transgranular fractures became dominant which consumed more
energy than intergranular fractures and resulted in more straight fracture path and less rough
fracture surface at high loading rates as supported by Zhang and Zhao (2013). Due to the
increased number of activated micro-cracks at high loading rates and that absorbed more
energy when compared to a single macro crack, resulting in an increase in the fracture energy
as parallel to the findings by Dai and Xia (2013).
Table 4.4 Summary of the failure loads and the fracture toughness results and their average
with standard deviations
Temperature Loading rate 𝑷 Average of 𝑲 Average of 𝑲
𝒎𝒂𝒙 𝑰𝑪 𝑰𝑪
(°C) (mm/min) (kN) 𝑷 (kN) (MPa·m0.5) (MPa·m0.5)
𝒎𝒂𝒙
0.02 1.28 1.70
0.05 1.95 2.59
RT (25) 1.93±0.49 2.56±0.65
0.08 1.98 2.63
0.1 2.49 3.30
0.02 1.94 2.57
0.05 2.04 2.69
100 2.14±0.18 2.84±0.24
0.08 2.25 2.99
0.1 2.32 3.08
0.02 1.50 1.99
175 0.05 1.65 1.88±0.34 2.19 2.49±0.45
0.08 2.10 2.79
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0.1 2.20 2.92
0.02 1.67 2.20
0.05 1.84 2.45
250 1.88±0.16 2.49±0.21
0.08 2.01 2.66
0.1 2.01 2.65
(a) (b)
Figure 4.12 (a) Pure mode-I fracture toughness variation with temperature (b) relationship of
mode-I fracture toughness with loading rate under different temperatures
In addition, the quasi-static mode I fracture toughness and energy-release rate of pre-heated
Australian granite are dependent on temperature as depicted in Figure 4.12. Under the same
loading rate, 𝐾 and 𝐺 of granite presented a decreasing trend by a total of approximately
𝐼𝐶 𝐼
17% and 30%, respectively with ascending temperature from ambient temperature (25 °C) to
250 °C. The fundamental reason for the decrease of fracture toughness is micro-cracks induced
by thermal damage resulting in degradation of the tensile stress resistance which indicates that
the rock’s ability to resist fracture deteriorated with increasing temperature. These results were
interpreted with the support of microscopic observations of the micro-cracks within the
specimens along with the help of SEM analysis (see Figure 3.13 in Chapter 3.3.5). This is also
in accordance with the findings of Yin et al. (2012), Mahanta et al. (2016) and Feng et al.
(2017). Therefore, it is shown that both the loading rate and temperature have significant
influence on the quasi-static mode I fracture toughness and energy-release rate of granite in
this study. These findings of this investigation will be useful for better understanding of the
strain burst mechanism such as application of a combination of favourable measures for
thermal damage and loading rate during deep excavations over 1000 m.
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4.4 - Dynamic characteristics of strain burst in brittle rocks exposed to thermal
effect
Rock fracture in explosion, excavation and strain burst tends to occur at high loading rates of
about 104-106 MPa·m1/2/s (Zhang and Zhao 2014), which is close to the loading rates in SHPB
tests. Hence, the SHPB apparatus is suitable for investigating the dynamic responses in rock
during strain burst. To explore the topic of coupled influence of thermal damage and loading
rate on the dynamic fracture properties and behaviour of Australian granite during strain burst,
a series of dynamic fracture toughness tests was conducted on thermally-treated CCNSCB
specimens over a wide range of loading rates by the SHPB setup. The dynamic mechanical
behaviour of granite after high-temperature treatment under different loading rates was
examined and discussed. The dynamic stress intensity factor (SIF) of the CCNSCB specimen
was obtained by the extended quasi-static calculation under the dynamic force equilibrium
condition. The dynamic initiation fracture toughness (DIFT) (𝐾𝑖 ) and the rate dependency of
𝐼𝑑
the phenomenon were determined and also compared for the specimens exposed to different
temperatures. The fracturing processes were recorded by a high-speed (HS) camera, and the
crack propagation speeds were estimated by HS image analysis. In addition, the dynamic
fracture process and the coupled influence of temperature and loading rate on the dynamic
fracture modes were identified by HS image analysis.
4.4.1 - Split Hopkinson Pressure Bar (SHPB) system
Dynamic fracture tests were performed by means of a 50 mm-diameter SHPB system at
Monash University as shown in Figure 4.13. The testing system comprises of a gas gun
generating the impact speed of the bullet up to 15 m/s, a cylindrical striker bar (500 mm in
length), an incident bar (2500 mm in length), a transmission bar (2000 mm in length) and an
absorbed bar (damper) (1000 in length), and were made from 50 mm diameter high strength
45CrMo steel, with a nominal yield strength of 1.1 GPa. The main parameters of the SHPB
setup used in this research are shown in Table 4.5. A steel platen with two pins was introduced
to achieve a three-point bending load to the specimen (see Figure 4.13).
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Table 4.5 The main parameters of the SHPB system ( Subscript b stands for bar)
Incident
Diameter Transmission Absorbing P-wave Elastic Density
bar
of bars bar length bar length velocity modulus 𝝆
length 𝒃
(mm) (mm) (mm) 𝒄 (m/s) 𝑬 (GPa) (kg/m3)
(mm) 𝒃 𝒃
50 2500 2000 1000 5170 210 7800
During the tests, the stress-wave pulses were captured by two sets of strain gauges located
diametrically opponent attached on the incident and transmission bars. An eight-channel digital
oscilloscope was used to record and store the strain gauge signals collected from the
Wheatstone bridge circuits after amplification (by means of a differential amplifier), together
with the signal from the strain gauge mounted on the CCNSCB specimen. The CCNSCB
specimen was sandwiched between the incident and transmission bars, with three point-
contacts to transfer dynamic loads: one between the incident bar and the top of the specimen,
the other two contacts formed by two supporting pins between the transmission bar and the
specimen, as depicted in Figure 4.13. To capture the fracture characteristics of Australian
granite under dynamic loading, a high-speed camera (CMOS camera, Phantom V2511) at the
frame rate of 200,000 fps with a resolution of 256 × 256 pixels in conjunction with the SHPB
system, located on the front side of the specimen, was utilised in this research (see Figure 4.13).
The focus of the ultra-high speed camera was manually adjusted under focused mode to capture
images with optimal quality.
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a part of the incident stress wave is reflected back into the incident bar as the reflected wave 𝜀
𝑟
upon reaching the bar-specimen interface, and the remaining portion of the wave passes
through the specimen to the transmission bar and becomes the transmitted wave 𝜀 . Strain
𝑡
gauges mounted on the incident and transmission bar surfaces capture the time of passage and
magnitude of these elastic stress-wave pulses through the incident and transmission bars during
the test.
Figure 4.14. The x-t diagram of stress waves propagation in SHPB (Xia et al. 2011)
Denoting the incident wave, the reflected wave and the transmitted wave by 𝜀 , 𝜀 and 𝜀 ,
𝑖 𝑟 𝑡
respectively, and based on one-dimensional elastic wave theory with the SHPB experimental
data the dynamic forces on the incident end (𝑃 ) and the transmitted end (𝑃 ) of the specimen
1 2
can be calculated as (Kolsky 1953) (see Figure 4.9b):
𝑃 = 𝐴 𝐸 (𝜀 +𝜀 ) (4.22)
1 𝑏 𝑏 𝑖 𝑟
𝑃 = 𝐴 𝐸 𝜀 (4.23)
2 𝑏 𝑏 𝑡
where 𝐴 and 𝐸 the cross-sectional area and Young’s modulus of the bars, respectively.
𝑏 𝑏
The histories of strain rate 𝜀̇(𝑡), strain 𝜀(𝑡) and stress 𝜎(𝑡) of the specimen in the dynamic tests
can be determined as:
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𝐶
𝜀̇(𝑡) = (𝜀 −𝜀 −𝜀 ) (4.24)
𝐿 𝑖 𝑟 𝑡
0
𝐶 𝑡
𝜀(𝑡) = ∫ (𝜀 −𝜀 −𝜀 )𝑑𝑡 (4.25)
𝐿 𝑖 𝑟 𝑡
0 0
𝐴
𝑏
𝜎(𝑡) = 𝐸 (𝜀 −𝜀 −𝜀 ) (4.26)
2𝐴 𝑏 𝑖 𝑟 𝑡
0
where 𝐴 , and 𝐿 are the initial cross-sectional area and the initial length of the specimen,
0 0
respectively. C is the one dimensional longitudinal elastic stress wave velocity of the bar.
Therefore, based on the Equations 4.24-4.26, the dynamic stress-strain curve of the specimen
can be determined.
4.3.2.2 - Pulse shaping technique
The induced stress wave is an approximately trapezoidal shape accompanied by high-frequency
oscillation and a steep rise of the incident wave when the striker bar directly impacted on the
incident bar. Without a proper pulse shaping, it is difficult to achieve dynamic stress
equilibrium which leads to premature failure of rock and unbalanced forces at the front and
rear interface of the rock sample (Zhou et al. 2012). In order to eliminate this problem, the
pulse shaping technique was adopted to facilitate the dynamic force balance of the CCNSCB
specimen which is a requirement for all the equations deduced in the SHPB test in this study.
4.4.2 - Dynamic fracture tests
The damage evolution of Australian granite was investigated by conducting dynamic tests over
a wide range of loading rates to reveal the rate dependency of strain burst. Dynamic fracture
toughness tests were performed on thermally-treated granite specimens up to 250 °C under
different impact velocity ranging from 2 to 8 m/s using a SHPB device at Monash University.
4.4.3 - Evaluation of the experimental results
CCNSCB granite specimens were successfully tested for dynamic fracture toughness
mechanical behaviour in the SPHB experiments. For all the SHPB tests, the dynamic force
balance of the granite specimen is inspected,and the results meet the criterion recommended
by the ISRM (Zhou et al. 2012). The influence of temperature and rate dependence of the
dynamic fracture toughness of Australian granite are analysed and discussed. The dynamic
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fracturing process and failure patterns of CCNSCB samples in different temperature and
loading rate conditions are observed using a high-speed camera.
4.3.2.1 - Dynamic force balance
Dynamic force equilibrium is the prerequisite of any effective dynamic fracture tests. It must
be ensured that the time-varying dynamic forces on both loading sides of the specimen are
roughly balanced prior to failure and the sample must be in a state of stress equilibrium through
the time to fracture and thus the quasi-static equation could be employed to determine the
dynamic fracture toughness. According to the suggested method by ISRM, the dynamic force
equilibrium was achieved for each sample by means of the pulse shaping technique in this
research (Zhou et al. 2012). Taking a typical test as an example, the captured incident, reflected
and transmitted strain waveforms of a typical CCNSCB sample are displayed in Figure 4.15a.
The time-zero of the incident and reflected waves was shifted to the incident bar/specimen
interface, and the time-zero of the transmitted wave was shifted to the transmitted bar/specimen
interface. As shown in Figure 4.15b, the curve of the sum of the incident and reflected stresses
almost overlapped (𝑃 = 𝑃 ) with that of the transmitted stress, indicating that the external
1 2
forces on both sides of the sample was nearly identical. The dynamic forces 𝑃 and 𝑃 were
1 2
calculated and checked by equations 4.22 and 4.23, and the dynamic loading history on both
ends of a specimen is shown in Figure 4.15b. It can be observed that the uniformity of the
dynamic stress across the specimen was well achieved in the impact direction, and thus the
inertial effect was reduced to a negligible level. Although there exists inevitably dynamic
friction at the interfaces between the rock sample and the bars, the achieved dynamic stress
equilibrium also demonstrated that 1D stress wave propagation theory could be employed to
calculate the stress-strain history of rock specimen in dynamic tests.
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(a) (b)
Figure 4.15 (a) Typical signals recorded by strain gauges of a dynamic test with thermally-
treated (100 °C) CCNSCB specimen at 𝑣 of 5 m/s and (b) dynamic force equilibrium.
𝑠𝑡𝑟𝑖𝑘𝑒𝑟
In., Re., Tr. denote the incident, reflected and transmitted waves, respectively
4.3.2.2 - Dynamic data interpretation
Figure 4.16a presents a typical dynamic stress-strain curve of a granite specimen in dynamic
CCNSCB test. The stress and strain were calculated from the incident and transmission bar
signals using Equations 4.25 and 4.26. These signals provide not only the deformation
information of the specimen, but also contain energy release during rock fracturing. The
evolution of stress and strain on the rock specimen during impact are shown in Figure 4.16b.
It should be noted that the stress of the peak point can be used to calibrate dynamic constitutive
models. Figure 4.16c depicts a typical dynamic SIF-time history curve of CCNSCB specimen
which can be used for determining the loading rate in the dynamic experiment.
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Using the signals of the incident, reflected and transmitted waves recorded by the strain gauges,
the stress-strain curves of granite were obtained under the coupling effects of temperature (25,
100, 175 and 250 °C) and impact velocity, 𝑣 , (2, 3, 5, 7 and 8 m/s), as presented in Figure
𝑠𝑡𝑟𝑖𝑘𝑒𝑟
4.17. It can be seen that the curves underwent into three stages: elastic deformation, yielding
and failure. In the elastic deformation stage, the rate of increase in the stress decreased more
slowly compared with that in the initial loading. Meanwhile, the micro-cracks within the rock
began to increase in size under the action of the dynamic loading, resulting in a decrease in the
curve slope. In the yielding stage, the rate of increase in the stress was lower than that in the
elastic stage, mainly due to the rapid expansion of the micro-cracks within the specimen unde
the stress wave. When the curve reached the peak strength, the maximum load-bearing capacity
was reached, which would led to macroscopic damage. In the failure stage, due to the formation
of macroscopic fracture surfaces the failure of rock occurred which resulted in the decrease in
the load-bearing capacity of the specimen. The stress decreased, while the strain continued to
increase in this stage. With an increase in the impact velocity, the loading rate strengthening
influence became more remarkable and the stress of the granite increased under all
temperatures. At a high impact velocity, the loading was fast and plastic strain component may
not get enough time to develop fully until the next incremental load was applied. Consequently
it appeared that the material had stiffened due to the incomplete development of the plastic
strain which then led to the increase of the dynamic strength of granite.
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Figure 4.17. Dynamic stress-strain curves of granite under different temperatures and impact
loadings
Figure 4.18 presents the relationship between the dynamic strength and the loading rate under
various temperatures. It can be seen that the loading rate has a significant effect on the dynamic
strength of granite under each temperature level, however the degree of the influence varies.
At a given loading rate or impact velocity, the value of dynamic strength for the same level of
deformation tended to decrease as the pre-heating temperature rose over the range from room
temperature (25 °C) to 250 °C due to degradation influence of thermal damage on the overall
rock strength in which high temperature aggravated the cumulative damage of the rock. Similar
results were observed by Yin et al. (2012) and Wang et al. (2018) who studied the mechanical
properties of granite by conducting dynamic tests using the SHPB technique. Taking 𝑣 =
𝑠𝑡𝑟𝑖𝑘𝑒𝑟
5 𝑚/𝑠 as an example, the dynamic strength of granite showed a decline by 33% when the
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et al. 2012), the loading rate (𝐾̇ ) of CCNSCB specimen was calculated by the evolution of the
𝐼
dynamic SIF obtained from the dynamic CCNSCB test. Figure 4.16c shows a typical dynamic
SIF-time history curve of CCNSCB specimen. There exists an approximately linear-increasing
regime in the SIF history, indicating the dynamic SIF in the CCNSCB specimen increased
steadily during this stage. The slope of this region is defined as the loading rate in which the
unit of the loading rate is GPa·m1/2 s-1 based on the suggested method by ISRM (Zhou et al.
2012). In this study, the loading rates of all specimens in dynamic CCNSCB tests were
determined using this method. Typical dynamic SIF-time curves including the loading rate in
the CCNSCB specimens at room temperature (25 °C) are depicted in Figure 4.19.
Figure 4.19. Typical SIF-time curves for determining loading rate in dynamic CCNSCB tests
at room temperature (25 °C)
4.3.2.4 - Thermal damage influence and rate dependence of dynamic initiation
fracture toughness (𝑲𝒊 )
𝑰𝒅
The dynamic initiation fracture toughness (DIFT) (𝐾𝑖 ) which is the ability of the material to
𝐼𝑑
fracture was determined by using the maximum value of SIF in this research. The fracture
properties were deduced using a quasi-static theory as the dynamic stress balance was
substantially achieved during the dynamic test using pulse shaping technique, eliminating the
inertial effects (Chen et al. 2009; Zhou et al. 2012). The DIFT of CCNSCB specimen was
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calculated by using Equations 4.2 and 4.3, provided that the dynamic force balance was
satisfied at both ends of specimens. Figure 4.20 shows the variation of DIFT with the loading
rate and temperature. It can be concluded from Figure 4.20 that the DIFT of granite is obviously
both loading rate and temperature dependent. The DIFT are close to each other at lower loading
rates (less than 400 GPa·m1/2 s-1), whereas, showed a certain degree of dispersion at higher
loading rates. For the CCNSCB specimen under the same loading rate, the DIFT values of
granite showed a decline compared with those at 25 °C. The obtained DIFT values of thermally-
treated granite under various impact velocities from dynamic CCNSCB tests are listed in Table
4.6. For instance, the DIFT under the impact velocity of 5 m/s, decreased by 29% as the
temperature increased from 25 °C to 250 °C. This phenomenon was mainly caused by the
increase of the thermal damage induced by the micro-cracks which eventually led to the
continuous decrease of fracture toughness. This viewpoint was further verified with the SEM
analysis conducted to observe the microstructure of the granite after treatment at various
temperatures in Chapter 3.
Figure 4.20. The DIFT versus loading rate for granite specimens treated at various
temperatures
In order to systematically investigate the coupling effects of loading rate and thermal damage
on the DIFT of granite, the linear regression method was utilised and the linear fitting of each
group was obtained. Figure 4.21 presents the rate dependency of DIFT for four groups of
thermally-treated granite. It was found that the DIFT of granite showed an increasing trend
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4.3.2.5 - Dynamic fracturing process and failure patterns of CCNSCB specimens
To study the progressive dynamic failure of thermally treated granite, a high-speed (HS)
camera with 200,000 fps was utilised to capture the dynamic fracturing process in dynamic
tests. The representative examples of typical dynamic mode I failure processes of CCNSCB
granite specimens induced by different temperature conditions at impact velocity of 8 m/s are
depicted in Figure 4.22, demonstrating the initiation and propagation of the cracks. The time
zero corresponds to a specific time when the incident pulse has just arrived at the incident
bar/specimen interface. The first one or two snapshots exhibit the typical CCSCNB specimen
prior to macro fracture onset. It can be seen that the cracks initiated from the tip of notch and
propagated along the impact loading, and then the tensile failure along the dynamic loading
direction dominated the failure. For instance, after around 154 μs, a small macroscopic crack
ahead of the notch tip became visible, indicating that crack initiation occurred, and then the
crack propagated along the pre-notched direction. Subsequently, the primary crack run
throughout the specimen at about 189 μs, and the CCNSCB specimen was split into two almost
identical halves and each fragment showed a rotation motion around the contact points between
the incident bar and the sample (see Figure 4.22, the last snapshot).
0 μs 154 μs 165 μs
LD
crack
173 μs 189 μs 218 μs
(a) T = 25 °C
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0 μs 59 μs 66 μs
LD
crack
72 μs 78 μs 132 μs
(d) T = 250 °C
Figure 4.22. HS camera images showing dynamic fracturing process of thermally treated
granite (a) 25 °C (RT) and (b) 250 °C at an impact velocity of 8 m/s in dynamic CCNSCB
tests (LD-loading direction)
The failure mechanism of rocks can be revealed by assessing the failure mode. The failure
patterns of Australian granite exposed to various temperatures at five different impact
velocities can be seen in Figure 4.23. Along with the increased impact velocity, the failure
modes of the pre-heated granite changed from tensile splitting (characterisation of class I) to
pulverisation in which the samples were pulverised by excess energy in class II loading,
indicating that the stress concentration at both ends became more serious, and thus the crashed
area was greater. The fundamental reason for this failure mode was that the elastic modulus of
the bar was quite different from that of the specimen, resulting in that the pressure of contact
surface was concentrated and thus the specimens were broken into many smaller fragments or
pulverised in which more cracks were activated and expanded.
It can be seen in Figure 4.23 that the increased level of thermal damage within the specimen
resulted in a wider damage zone which was due to the thermally-induced micro-cracks with
the treatment temperature. This can be attributed to the weakening of the minerals’ bonding
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25 °C 100 °C 175 °C 250 °C
(e) Impact velocity (𝑣 ) = 8 m/s
𝑠𝑡𝑟𝑖𝑘𝑒𝑟
Figure 4.23. Failure modes of recovered specimens under different impact velocities and
temperatures
4.5 -Summary and discussion
In this chapter, the effects of the thermal damage and loading rate on both quasi-static and
dynamic mechanical, fracture characteristics and quasi-static (𝐾 ) and dynamic initiation (𝐾𝑖 )
𝐼𝐶 𝐼𝑑
mode I fracture toughness and energy-release rate of thermally treated Australian granite
specimens at various pre-heating treatments up to 250 °C under different loading rates were
explored. The CCNSCB specimens were adopted in the quasi-static and dynamic mode I
fracture toughness measurements of the rocks. A servo-hydraulic testing machine and a
dynamic testing apparatus SHPB were utilised to conduct the quasi-static and dynamic fracture
toughness tests. The fracturing characteristics during strain burst under various temperature
conditions and loading rates were assessed and discussed in detail. The following key
conclusions can be drawn:
1. The CCNSCB specimen combines the merits of two ISRM-suggested methods (CCNBD
and NSCB methods), and thus it allows accurate determination of the mode I fracture
toughness of granite under quasi-static and dynamic loadings.
2. The experimental results indicated that the quasi-static fracture toughness and energy-
release rate in mode I are a function of loading rate and they presented a rising trend with
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increasing loading rate. At high loading rates, transgranular fractures became dominant
which consumed more energy than intergranular fractures; this in turn, resulted in more
straight fracture path and posed a less rough fracture surface when compared to the low
loading rate condition (Zhang and Zhao 2013).
3. Under the same loading rate, the quasi-static mode I fracture toughness and energy-release
rate of granite showed a gradual decrease (17% and 30%, respectively) with ascending
temperature from 25 °C to 250 °C due to the thermally-induced micro-cracks within the
rocks. These findings of this investigation will be useful in achieving a better
understanding of initiation of fracturing during strain burst under various temperature and
loading rate conditions.
4. The stress-strain curves of granite under various impact velocities and temperatures
showed the same deformation stages; elastic deformation, yielding and failure. When the
impact velocity was high, the loading rate strengthening effect became more remarkable
and the strength of granite increased under all temperatures. The failure modes of
Australian granite also exhibited rate dependence at the same temperature level. Along
with the high impact velocity, the failure mode of the pre-heated granite changed from
tensile splitting (characterisation of Class I) to pulverisation or breaking into many small
pieces in which the specimens were pulverised by the excess energy in Class II loading.
Under the same dynamic impact, an increase in the treatment temperature weakened the
interaction force between the particles and aggravated the fragmentation degree of granite.
5. The DIFT of Australian granite was obtained by the quasi-static analysis that was
evidenced by the dynamic force balance until the time to fracture. The DIFT of the granite
presented an ascending trend with the loading rate at a given heat-treatment temperature
and decreased with increasing temperature, revealing the deterioration of the ability to
resist fracturing with the rise of temperature. Therefore, in order to effectively and safely
excavate the rock in deep underground conditions, a favourable measure should be applied
to reduce the intensity of strain burst by considering a combined application of thermal
treatment and impact with a proper loading rate.
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Chapter 5: Effects of thermal damage on strain burst
mechanism for brittle rocks under true-triaxial loading
conditions
Strain burst is a common problem encountered in brittle rocks in deep, high-stress mining
applications. Limited research focuses on the effects of temperature on the strain burst
mechanism and the kinetic energies of rocks. This study aims to investigate the effects of
thermal damage on the strain burst characteristics of brittle rocks under true-triaxial loading
conditions using the acoustic emission (AE) and kinetic energy (KE) analyses. The Time-
domain and frequency-domain analyses related to strain burst were studied, and the damage
evolution was quantified by b-values, cumulative AE energy and events rates. The ejection
velocities of the rock fragments from the free face of the granite specimens were used to
calculate kinetic energies. The experimental results showed that thermal damage resulted in a
delay in bursting but increased the bursting rate at ~95% of normalised stress level. This is
believed to be due to the microcracks induced by temperature exposure and thus the
accumulated AE energy (also supported by cumulative AE counts) at the initial loading stage
was reduced, causing a delay in bursting. The strain burst stress, initial rock fragment ejection
velocity, and kinetic energy decreased from room temperature (25 °C) to 100 °C, whereas they
resulted in a gradual rise from 100 °C to 150 °C demonstrating more intense strain burst
behaviour.
Keywords Strain burst · Rock burst · True-triaxial loading · Thermal damage · Temperature
· Acoustic emission · b-value · Kinetic Energy
Rock burst is a typical unstable rock failure associated with the violent ejections of rock
fragments from the free face/sidewall/roof of an underground excavation. A serious threat, rock
bursts can kill workers and cause severe injuries and damage. They can also cause mining and
tunnelling operations to cease either temporarily or permanently. Rock bursts are classified into
three types: Strain burst, fault-slip burst, and pillar burst (Hedley 1992). Strain burst is the most
prevalent type of rock burst. It occurs due to the sudden release of stored strain energy within
the rock mass when the induced major principal stress (σ ) exceeds the rock mass strength
1
(σ ). This type of detrimental failure process has been observed in deep, hard rock mines and
cm
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conditions
tunnels in different locations all around the world, and is considered to be the biggest unsolved
problem in deep underground excavations (He et al. 2016). Underground rock mass is in a state
of stress equilibrium prior to any excavation (σ >σ >σ ). Introducing an excavation in rock
1 2 3
masses results in the redistribution of stresses around underground openings (see Figure 5.1)
and accumulation of elastic strain energy in the surrounding rock mass.
Figure 5.1 Stress state change on the sidewall of an underground opening, and a
representative elementary volume before and after excavation (“modified from Su et al.
2017a”)
Additionally, rock mass surrounding underground excavations is vulnerable to the effects of
high ground temperatures, especially at increasing depths. The physical and mechanical
behaviours of the rock mass are influenced by the thermal effects which threaten both the safety
of the working environment and the efficiency of engineering projects (Chen et al. 2012; Liu
and Xu 2013). For instance, a number of intense strain bursts occurred during the excavation
of tunnels in the Jinping II Hydropower Station, which caused casualties and fatal injuries,
damaged equipment and ceased operations at the increasing depth due to the high geo-stress
and high temperature (Zhang et al. 2012; Li et al. 2012; Feng et al. 2015). Understanding
thermally induced rock damage is, therefore, of utmost importance for the safety and long-term
stability of underground excavations. For this purpose, a realistic experimental testing system
needs to be used for the assessment of thermal damage on the behaviour of strain burst.
Many researchers have investigated the influence of temperature on the mechanical and
physical behaviour of rocks under uniaxial compression (Heuze 1983; Dwivedi et al. 2008;
Sun et al. 2015), and under triaxial compression (Masri et al. 2014; Ding et al. 2016; Yao et al.
2016; Mohamadi and Wan 2016). Ding et al. (2016), studied damage evolution in sandstone
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conditions
after exposure to high-temperature treatment in unloading conditions, and found that both peak
ductile deformation and peak effective stress changed after a critical temperature level. Kong
et al. (2016) investigated the AE characteristics and physical-mechanical properties of
sandstone after high-temperature exposure under uniaxial compression conditions and found
that AE parameters can be used for evaluating the thermal stability of rocks and for analysing
crack development. These existing works clearly show considerable thermal effects on the
mechanical behaviour of rocks, and the need to consider damage due to thermal effects in
investigating strain burst in deep mining. In this sense, a true-triaxial condition that better
reflect stress states in deep mining, along with the effects of thermal damage on strain burst
behaviour of rocks should be considered. However, to the best of our knowledge, all these
features are either missing or not addressed at length in previous works.
A considerable amount of research in the laboratory has been conducted to mimic the failure
process of strain burst. These experimental efforts have mainly conducted under uniaxial
compression (Nemat-Nasser and Horii 1982; Wang and Park 2001), conventional triaxial
compression (Huang et al. 2001; Hua and You 2001;), and true-triaxial compression (Mogi
1971; Atkinson and Ko 1973; Michelis 1985; Takahaski and Koide 1989; Wawersik et al. 1997;
Haimson and Chang 2000; Nasseri et al. 2014; Feng et al. 2016). However, none of the
aforementioned testing methods were able to realistically simulate the exact boundary
conditions and stress paths for rocks during an excavation in which strain burst occurs. Hence,
to characterise strain burst process in the laboratory, a novel true-triaxial strain burst testing
system was developed by He et al. (2010) at the State Key Laboratory for Geomechanics and
Deep Underground Engineering in Beijing, China. This hydraulic testing facility enables
researchers to simulate the creation of an excavation by abruptly unloading σ from one of the
3
rectangular prism’s surfaces that is exposed to air. Using this testing system, a considerable
number of tests have been conducted on various types of rocks exposed to different stress paths
to provide a better understanding of the behaviour of strain burst under true-triaxial
loading/unloading conditions (He et al. 2010, 2012, Gong et al. 2015; Li et al. 2015). Few
studies in the available literature have addressed the kinetic energy characteristics of strain
burst failure. The influence of the unloading rate on strain burst behaviours of brittle rock under
true-triaxial unloading conditions was studied by Zhao et al. (2014) concluding that the rock
tends to strain burst more often when the unloading rate is high and the failure mode changes
from strain burst to non-violent spalling as the unloading rate decreases. After creating a
comprehensive database on the true-triaxial unloading tests, Akdag et al. (2017) discussed the
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conditions
influence of specimen dimensions on the bursting behaviour of rocks and indicated that the
failure mode changes from strain bursting to local spalling when the height to width ratio of
the rock sample is reduced from 2.5 to 1. For this reason and my focus on rock burst in the
present study, all specimens with height to width ratio of 2.5 were used. Su et al. (2017)
investigated the influence of tunnel axis stress on strain burst by using modified true-triaxial
rock burst system. The experimental results indicated that intensive strain burst is more likely
to occur when the tunnel axis stress is high. Table 1 summarises the true-triaxial loading and
unloading tests to assess the failure characteristics of different rocks. However, the
aforementioned studies did not consider the temperature influence on strain burst behaviours.
Therefore, it is essential to investigate how strain burst mechanism is affected by high-
temperature conditions.
This chapter investigates the influence of temperature on strain burst. A true-triaxial loading-
unloading experimental set up was used to replicate strain-burst condition. In the following
sections, the basic properties of the rock samples are described first. The strain burst testing
methods and the experimental procedure are then introduced. This is followed by the analysis
of the influence of temperature on strain burst stress and dynamic failure processes of strain
burst. Subsequently, time-domain, frequency-domain and b-value analyses were conducted to
systematically investigate the evolution of AE due to thermal damage influence on strain burst.
Finally, the kinetic energies of the ejected rock fragments due to thermal damage are discussed.
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1 Table 5.1 Summary of true-triaxial loading and unloading tests to characterise the failure type of rocks
Loading Specimen size
Loading method Rock type Failure mode Reference
type (mm x mm x mm)
15 x 15 x 30 Dolomite Mogi (1971)
50 x 50 x 100 Marble Michelis (1985)
(1) apply σ , σ , σ 50 x 50 x 100 Sedimentary rocks Takahashi & Koide (1989)
1 2 3
Loading (2) keep σ and σ 57 x 57 x 125 Sandstone Fracturing & ductility Wawersik et al. (1997)
2 3
(3) increase σ 1 19 x 19 x 38 Granite Haimson & Chang (2000)
80 x 80 x 80 Sandstone Nasseri et al. (2014)
50 x 50 x 100 Granite Feng et al. (2016)
Limestone, granite,
30 x 60 x 150 Rock burst He et al. (2010, 2012)
sandstone, marble
20 x 40 x 100 Marble Spalling Coli et al. (2010)
30 x 60 x 150 Marble Rock burst and slabbing Gong et al. (2012)
(1) apply σ , σ , σ
1 2 3
(2) keep σ 30 x 60 x 150 Granite Rock burst Zhao et al. (2014)
Unloading 2
(3) Unload σ 30 x 60 x 150 Granite Rock burst
3
(4) Increase σ 1 30 x 60 x 120 Granite Slabbing Zhao and Cai (2014)
30 x 60 x 90 Granite Shearing
Granite, sandstone, Splitting, Slabbing,
100 x 100 x 100 Li et al. (2015)
cement mortar Spalling
100 x 100 x 200 Granite Rock burst Su et al. (2017)
25 x 50 x 125 Granite Strain burst Akdag et al. (2018)
2
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5.1 -Experimental methodology
5.1.1 - Rock properties
The rock samples used in this study were collected from a borehole located in South Australia
at a depth of 1020 – 1345 m. The collected rock was coarse-grained granite with weak to
moderate alteration and occasional weak gneissic foliation. The grain size of this brittle granite
rock ranges from 0.5 mm to 3 mm and is composed of potassium feldspar, quartz and chlorite.
Therefore, the diameter of the specimens was more than 10 times bigger than the rock grain
size required to satisfy ISRM recommendations (Fairhurst and Hudson 1999).
Uniaxial compression tests were performed on both cylindrical granite specimens that had a
diameter of 42 mm, were sub-cored from 63 mm diameter drill cores, and were 100 mm long
(Fairhurst and Hudson 1999). The tests were also performed on rectangular prism samples (125
mm × 50 mm × 25 mm). The granite specimens were loaded axially with an axial displacement
rate of 0.1 mm/min and LVDTs and strain gauges were attached to measure both axial and
lateral strains. Rocks were also equipped with AE sensor to capture the cracking and damage
behaviour during the tests (see Figure 5.2). The test results and basic mechanical properties of
the granite samples are listed in Table 5.2.
Figure 5.2 Instrumentation of granite specimens for UCS tests
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Table 5.2 Mechanical properties of rectangular prism granite specimens for UCS (𝜎 ) tests
𝑐2
Dimensions Young's
Specimen Density UCS,𝝈 Poisson's
𝒄𝟐
Height modulus,
Number Width (mm) Thickness (mm) (g/cm3) (MPa) ratio, ν
(mm) E (GPa)
B1 #5 124.87 50.10 25.02 2.89 175.8 55.3 0.19
B1 #8 124.99 50.23 25.14 2.82 184.4 27.9 0.11
B3 #3 125.04 49.97 25.00 2.87 137.1 28.5 0.10
5.2 - Experimental procedure for strain burst tests
5.2.1 - Sample preparation and strain burst testing system
A total of sixteen rectangular prism granite samples were prepared from the drill cores of 63
mm diameter for the strain burst tests (see Figure 5.3a). Each sample size was approximately
125 mm × 50 mm × 25 mm. All six surfaces of the samples were carefully polished to minimise
the end effect during loading. The samples’ average flatness was 0.009 mm. Nine flatness
measurements were taken from the surfaces of each specimen using digital dial gauge. Sample
hardness was measured with the Leeb rebound method, using an Equotip 3 hardness tester (see
Figure 5.3b-c). The Leeb number (L value) is used to express the hardness of the material,
which can be used as an indicator of rock strength (Aoki and Matsukara, 2008). The average
Leeb hardness of the granite specimens used for this study was 746 and the average density
was 2871 kg/m3. The average P-wave velocity of the specimens before thermal damage was
approximately 5764 m/s. All the granite specimens were divided into six groups (i.e. groups I,
II, III, IV, V and VI) based on temperature. Specimens were then kept at room temperature of
25 °C (i.e. group I) or heated up to the following temperature levels of 50, 75, 100, 125, and
150 °C (i.e. groups II, III, IV, V and VI respectively).
Figure 5.3 (a) Overview of granite specimens, (b) flatness measurement by digital dial gauge,
(c) hardness measurement via Equotip hardness tester
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The strain bursts tests were performed using the deep underground true-triaxial strain burst
testing system developed by He et al. (2010) at the University of Mining and Technology in
Beijing, China. The strain burst test facility consists of a hydraulic controlling unit, a data
acquisition system for stress and deformation, and also equipped with an AE monitoring
system, a high-speed digital video camera system to monitor the instantaneous strain bursting
process and linear variable differential transducers (LVDT) to measure the displacements
during testing (see Figure 5.4). To mimic and characterise the stress distribution near an
excavation boundary in the laboratory, this testing system enables loading a rectangular rock
specimen independently in three principal stress directions (σ , σ , σ ) progressively to the pre-
1 2 3
determined in-situ stress level, and suddenly removing σ by dropping a rigid loading plate,
3
while maintaining σ constant and then increasing σ until strain burst occurs (see Figure 5.4d-
2 1
e). The hydraulic loading unit has a maximum force capacity of 450 kN which is used to apply
vertical and horizontal loads on the six surfaces of a rectangular rock specimen. The data
acquisition system is capable of recording 100,000 data points per second (see Figure 5.4a).
The high-speed digital camera records at 1,000 fps with a resolution of 1024 × 1024 pixels,
which enables the capture of sudden cracking as well as the violent ejection of rock fragments
(see Figure 5.4e).
The AE technique is a useful, non-destructive testing method used to investigate the onset and
evolution of micro-cracking. It is also used to analyse the damage mechanism of rocks
(Karakus et al. 2016). In the present study, two AE sensors with a diameter of 18 mm to
investigate the AE characteristics of granite samples were used. The AE transducers (type WD,
from the American Physical Acoustics Corp.) were attached to the lateral side of the rock
specimens by means of spring clips and adhesive tape to minimise friction between the
specimen and the loading plate and to prevent sensor failure due to rock ejection (see Figure
5.4f). A petroleum jelly was smeared on the sensors and the steel plates to ensure good acoustic
coupling. The resonance frequency of the AE transducers was 125 kHz, associated with an
operating frequency range from 100 kHz to 1 MHz. A PCI-2 AE system was used to monitor
the damage within the granite specimens during strain burst tests and the output voltage of the
AE was amplified to 40 dB gain. The amplitude threshold for AE detection was set to 35 dB
with an AE sampling rate of 10 msps (million samples per second) for each test.
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Cai (2008) stated that it is significant to be able to capture the correct rock mass behaviour
during excavations, because the actual stress path in a rock mass is complex and has an
important role in the failure or damage process. Hence, accurate excavation responses depend
on the unloading paths. The in-situ stress test results were used as a guideline for determining
the stress loading conditions used to simulate strain burst in the laboratory. Figure 5.5 plots the
designed stress path and the applied loading-unloading directions on a rock specimen during
strain burst testing. All surfaces of the rectangular prism granite specimen were loaded
independently, in three principal stress directions. The loads were progressively applied until
all six surfaces reached the minimum principal stress. Subsequently, while the loads on two
surfaces, where 𝜎 was acting, were kept constant, the loads on the other four surfaces were
3
increased simultaneously until they reached the intermediate principal stress level. Finally,
while keeping the loads on the other lateral four surfaces constant, the load at the top surface
was increased to the pre-determined maximum principal stress level in two steps. Therefore,
the in-situ stress level of σ /σ /σ = 43/23/11 MPa was reached and the loads were retained for
1 2 3
about 5 minutes to make sure the stress was distributed uniformly. In order to mimic the stress
redistribution and concentration after an excavation, σ was removed quickly with an unloading
3
rate of around 17 MPa/s while σ was kept constant. Then to generate a strain burst σ was
2 1
increased at a constant rate of 0.25 MPa/s until strain burst occurred. Meanwhile, when
unloading of σ began, recording of the high-speed digital video camera was started to capture
3
the strain burst process.
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Figure 5.5 Designed loading-unloading stress path and illustration of stress conditions on
rock specimen for strain burst tests
5.3 -Evaluation of the experimental results
5.3.1 - Influence of thermal damage on strain burst stress
The principal stresses applied to the granite samples just before unloading, and at failure, under
various temperature conditions are summarised in Table 5.3. The table shows the ratios of
major principal stress σ , the sum of major and intermediate principal stresses, and the
1
deviatoric stress to the UCS (σ ,σ ) of both cylindrical and rectangular prism granite
c1 c2
specimens. Note that σ is the average value of UCS of cylindrical granite specimens (42 mm
c1
× 100 mm), which is equal to 155 MPa and σ corresponds to the average UCS value of
c2
rectangular prism specimens (25 mm × 50 mm × 125 mm), which is 180 MPa. The major
principal stress σ at failure varies in the range of 0.65–1.87 times σ , and 0.56–1.61 times
1 c1
σ . It is also shown that the ratio of deviatoric stress of σ and σ to σ and σ is between
c2 1 2 c1 c2
0.49–1.70 and 0.42–1.46 respectively. The ratios indicated in Table 5.3 can be used as
indicators of strain burst occurrence by comparing them to the rock burst criteria based on
strength theory. Figure 5.6 presents the actual stress paths and cumulative AE energy, which
was calculated after AE analysis, of the granite specimens from each group under different
temperature conditions. As the testing system was not servo-controlled, you will see in Table
3 some discrepancies can be conserved between the recorded principal stresses and the
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5.3.2 - Observations on the influence of thermal damage on strain burst
behaviour
In order to capture the failure processes of the granite samples induced by the different
temperature conditions, a high-speed camera was used. Using a frame rate of 1000 f/s (frames
per second), the dynamic failure characteristics of the tested samples, including the crack
growth and fragment ejection were observed. A series of images for the samples were captured
to investigate the influence of temperature on the rock failure process. These are presented in
Figure 5.9. The numbers at the bottom-left corner of the snapshots indicate time in h:m:s:ms.
It should be noted that regardless of the temperature, strain bursts occurred in all specimens. A
common strain burst development process for all of the specimens was as follows: Splitting of
rock into rock plates, bending of the rock plates, ejection of rock fragments, and rock plates at
high speeds accompanied by a loud explosion sound after the rock plates break off. It can be
observed from Figure 5.9 that the intensity of the strain burst differs moderately in different
temperature conditions. For granite specimen tested at the temperature of 25 °C, (see Figure
5.9a), where the specimen did not experience any thermal damage, the upper part of the free
face split into rock plates, and small fragments were ejected at high speed. After the upper rock
plate broke off, a large number of fragments and rock plates were suddenly ejected outward,
and this activity was associated with a loud sound. The final strain burst pit area was around
half of the whole free surface of the specimen and tensile cracks near the free face occurred
parallel to σ on both lateral sides. When the temperature was increased up to 100 °C (see
1
Figure 5.9d), strain burst further became less violent. This may be caused by the thermal
damage due to the deteriorated bonding among mineral grains that rendered the rock relatively
weaker after temperature. Tensile cracks are observable at the free face of the sample. As the
temperature increased from 100 °C to 150 °C, more violent strain burst characteristics were
observed, as shown in Figure 5.9e-f. This gradual change can be attributed to the compaction
of the rock samples due to the closure of pre-existing micro-cracks (Kumari et al. 2017a).
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Figure 5.9 Rock failure process of the granite specimens treated with different temperatures
captured by the high-speed camera: (a) T = 25 °C; (b) T = 50 °C; (c) T = 75 °C; (d) T = 100
°C; (e) T = 125 °C; (f) T = 150 °C
5.3.3 - AE analysis for thermal damage assessment
It is well understood that rock failure is accompanied by the release of energy. Elastic waves
propagating from a source within a material by the rapid release of localised energy can be
defined as an acoustic emission. The AE method has been widely used to investigate brittle
rock failure, and to quantify rock damage in many engineering applications (Lockner 1993,
Grosse and Ohtsu, 2008; Nicksiar and Martin 2012; Carpinteri et al. 2013; Zhao et al. 2015;
Karakus et al. 2016). As shown in Figure 5.4, the AE technique was used to monitor the
evolution of damage inside the granite samples at various temperatures.
Time-domain analysis
AE parameters such as counts, hits, energy, amplitude and frequency were obtained from the
AE monitoring system and the fracturing processes of strain burst under different temperature
conditions were investigated. While the number of cracks is manifested by AE hits, the
magnitude of the micro-cracking is related to the AE energy. Cumulative AE energy was
therefore used to assess the energy release characteristics of the granite specimens subjected to
various temperatures under true-triaxial unloading conditions. Figure 5.6 illustrates the
evolution of cumulative AE energies of the samples. It can be seen that although temperature
conditions were different, the evolution features of cumulative AE energy for the six specimens
underwent a similar trend from the beginning of loading until strain burst. Based on the
cumulative AE energy characteristics, the evolution of AE behaviour was divided into three
typical stages, i.e., the AE quiet linear elastic deformation stage, the AE growth stage and the
AE active strain burst stage. Figures 5.10a and 5.11a depict the rate and cumulative plots of
the AE energy and hits versus the time and also corresponding normalised strain burst stress in
which the three deformation stages of strain burst are also demonstrated. The damage caused
by temperature was quantified by changes in AE signal characteristics. Therefore, thermal
damage for strain burst (𝐷 ) can be calculated for the granite specimens treated with different
𝑆𝐵
temperature conditions by using Equation 5.1.
𝛺
𝐷 = (5.1)
𝑆𝐵 𝛺
𝑚
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conditions
Figure 5.11 Plots of (a) AE hits rate and (b) cumulative AE hits and damage evolution by AE
hits versus normalised strain burst peak stress at corresponding stages shown in part a for the
rock at temperature of 25 °C
At the initial stage, a sudden increase can be observed due to the closure of pre-existing cracks,
voids or other defects. After the majority of the natural cracks compacted, the rocks went into
a linear elastic deformation period. During the stress maintenance phase, the cumulative AE
energy rate changed little indicating that no micro-cracking inside the rocks was observed.
During this phase, stiffness started to decrease, and it was associated with signifying tensile or
shear movements between the faces of closing or closed cracks (Eberhardt et al. 1998). Upon
the unloading of the minimum principal stress σ , the cumulative AE energy gradually
3
increased, revealing that new micro-cracks generated and started to grow. However, their low
AE energy indicates that they have limited influence on decreasing the overall strength of the
rock and thus cannot cause strain bursting. As the maximum principal stress σ was further
1
increased while intermediate principal stress σ was maintained constant, the micro-cracks
2
began to propagate to a few large cracks, to coalescence and to form macro-cracks. This
increasingly contributed to the degradation of the inherent rock strength, which was revealed
by a high amount of cumulative AE energy. At AE active strain burst stage, due to the unstable
coalescence of macro-cracks and the ejection of rock fragments from the free face, cumulative
AE energy associated with higher amplitudes rapidly increased at a high rate until strain burst
occurred. Figure 5.12 presents variations on the cumulative AE energy and cumulative AE
counts with the temperature for all granite specimens. In general, increasing the number of
micro-fractures caused a decline in both cumulative AE energy and counts. Nevertheless, as
observed in this work, this trend is only correct for sufficiently high temperatures. For example,
when the temperatures reached 100 °C and 150 °C, the cumulative AE energy of the samples
decreased by 14%-20%, and the cumulative AE counts declined by 20%-55%, compared with
the values at 25 °C.
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conditions
Figure 5.14 Thermal damage influence on damage accumulation rate
b-value analysis
The b-value from Gutenberg-Richter’s equation (Gutenberg and Richter 1956) has been widely
used to assess the internal damage evolution of rock (Grosse and Ohtsu 2008; Carpinteri et al.
2009; Sagar et al. 2012; Kim et al. 2015). The Gutenberg-Richter relation between the
cumulative frequency-magnitude distributions of AE data is given in seismology by Equation.
5.2.
𝐴
𝑙𝑜𝑔 (𝑁) = 𝑎 −𝑏( 𝑑𝐵 ) (5.2)
10 20
where 𝐴 is the peak amplitude of AE events in decibels, N is the incremental frequency which
𝑑𝐵
can be defined as the number of AE hits with an amplitude greater than 𝐴 and the b-value is
𝑑𝐵
the negative slope of the log-linear plot between frequency and amplitude.
For three deformation stages, b-values were calculated by plotting the cumulative AE hits, peak
amplitude distribution, and fitting curve (an example of calculation of b-values can be seen in
Figure 5.15a). Fracture density can be represented by the y-intercept of the fitting line and as
can be observed that y-intercepts of the three deformation stages decrease from the initial AE
quiet stage to the AE active stage.
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Figure 5.15 Example of calculation of b-values (a) AE incremental frequency and amplitude
distribution and b-value calculation, (b) average b-values and standard deviations in three
deformation stages for the granite specimen at temperature level of 150 °C (c) temperature
influence on b-value at AE active stage
Figure 5.15b presents the estimated b-values in three deformation stages and at the evolution.
At the initial stage, the closure and compaction of pre-existing microcracks, voids or other
defects resulted in high b-values. This is evidenced by a large number of AE events with low
magnitude. During the generation of new micro-cracks, and also during the stable growth of
micro-cracks (no macro-crack formation), a few AE events were observed. In the AE active
stage, b-values decreased sharply. This indicates that AE events with higher amplitudes were
detected due to the accelerated unstable crack growth, and coalescence until strain burst. This
sudden change in the b-value also indicates that the damage accumulated inside the rock is
increasing. Therefore, the higher b-value trend suggests that micro-crack growth, whilst lower
b-value trend implies that macro-cracks have formed inside the rock that can be used as a
damage alert.
Figure 5.15c presents the influence of temperature on the b-value at AE active stage. Although,
Carpinteri et al. (2009) indicated that b-value changes systematically from 1.5 (in which
damage in the material is still uniform at a condition of criticality) to 1.0 when the final failure
is imminent characterised by a strong damage localisation, b-values in Figure 5.15c are less
than 1.0 since they were calculated for AE active stage. When the temperature increased to 100
°C, b-values show an increasing trend. This indicates that thermal damage reduced the macro-
cracking process due to the mechanical degradation of the samples which in turn resulted in
less intense strain bursting. As the temperature increased from 100 °C to 150 °C, b-values
gradually declined which can reveal more intense strain burst characteristics. Therefore, b-
value analysis can be used to assess the type of deterioration of the rock and to quantify the
damage degree.
Frequency-domain analysis
The frequency-amplitude characteristics of the AE waves of the six granite specimens treated
different temperatures are presented in Figure 5.16. The frequency-amplitude behaviours of the
AE signals showed trends similar to the total cumulative AE energy responses. Increasing the
temperature led to a low-frequency band of and higher amplitudes (see Figure 5.16). When the
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conditions
Figure 5.16 AE frequency-amplitude features of the six granite specimens treated with
different temperatures: (a) T = 25 °C; (b) T = 50 °C; (c) T = 75 °C; (d) T = 100 °C; (e) T =
125 °C; (f) T = 150 °C
In order to investigate the influence of thermal damage on strain burst behaviours in greater
depth, the frequency spectrum analysis was carried out. The AE signals were analysed using
the Fast Fourier Transform (FFT) method (see Equation 5.3), as the frequency spectrum can
be used to investigate the internal damage level during strain burst.
𝑁−1
𝑋 = ∑ 𝑥 .𝑒−𝑖2𝜋𝑘𝑛/𝑁 (5.3)
𝑘 𝑛
𝑘=0
Figure 5.17 demonstrates the main frequency behaviour when the temperature was increased
from room temperature (25 °C) to 150 °C. The average results show that the main frequency
was approximately 261 kHz for room temperature samples and continually decreased to around
113 kHz as the temperature was increased. It is believed that the micro-cracking processes
occurred over a long time period at low temperatures. However, when temperature increased,
this micro-cracking period gradually diminished due to the thermal damage inside the
specimens.
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conditions
The kinetic energy calculation analysis of the ejected fragments can be described as follows.
First, a three-dimensional spatial coordinate system was set up in which the centre bottom of
the steel rig was selected as the origin point, denoted by a red circle (see Figure 5.18a). Then,
the motion trail of relatively large fragments was traced after bursting, as illustrated in Figure
5.18b. The specific spatial locations of the fragments were determined from the side and top
view of the high-speed photos (see Figure 5.18d). Figure 5.18c presents, the movement tracking
of the fragment, F-2, from the free face of the granite sample at the onset of bursting to the
bottom platform. After calculating the movement time, ∆𝑡, locations of the fragments before
and after ejection were identified with respect to the spatial coordinate system. As can be seen
in Figure 5.18b, the initial ejection location of the fragment is point A (𝑥 ,𝑦 ,𝑧 ), which has
0 0 0
an initial speed of 𝑉 and the final dropping down point is point B (𝑥 ,𝑦 ,𝑧 ).
0 1 1 1
After measuring the velocity, the total kinetic energy of the ejected fragments was calculated
by using Equation 5.4.
𝑛
1
2
𝐸 = ∑ 𝑚 𝑣 (5.4)
𝑘 2 𝑖 𝑖
𝑖=1
where n is the number of fragments having D > 10 mm and m > 0.5 g, 𝑚 is the mass of the 𝑖th
𝑖
rock fragment and 𝑣̅ is the initial ejection velocity of the 𝑖th rock fragment. By using the
𝑖
equation above, the total kinetic energies for all granite specimens treated with different
temperatures were calculated. Note that average velocity values of the ejected fragments were
taken as the ejection velocity of a granite specimen. The ejection velocities and strain bursting
of the granite specimens exposed to different temperature conditions from room temperature
(25 °C) to 150 °C are displayed in Figure 5.19. Due to the thermal damage occurred inside the
granite samples leading to the degradation of the mechanical characteristics, the ejection
velocity of the fragments dramatically decreased when the temperature level was below 100
°C. With improved compactness between 100 °C and 150 °C, the velocity of the ejected
fragments increased slightly, which is associated with relatively intense strain bursting (see
Figure 5.20a).
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conditions
Figure 5.19 Ejection velocities of rock fragments from the granite specimens treated with
different temperature conditions
The kinetic energy of the ejected fragments showed a trend similar to the ejection velocities.
Kinetic energy continually decreased with the temperature, until the critical temperature level
of 100 °C was reached. This is because the granite specimens manifested thermal damage (see
Figure 5.20b). The strain burst stress and total elastic strain energy showed a decline in
temperatures below 100 °C due to thermally induced damage and is shown in Figure 5.20a. It
can also be seen that the amount of total elastic strain energy released from the granite
specimens decreased because the thermally induced microcracks reduced the amount of strain
energy accumulation (see Figure 5.21b). When the temperature increased from 100 °C to 150
°C, the accumulated strain energy within the granite specimens increased (see Figure 5.21a).
Therefore, this led to the higher amount of the strain energy release associated with an increase
in kinetic energy, as shown in Figure 5.20a.
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conditions
energy and the ejection velocity of the fragments decreased by 45%, 68%, 96%, and 82%
respectively. It is believed that thermally induced microcracking caused mechanical
degradation and this resulted in less strain energy accumulation which led to small kinetic
energy. When the temperature level was above 100 °C, bursting stress, accumulated strain
energy, kinetic energy release and fragment ejection velocity increased when compared to the
results captured at the temperature of 100 °C. This led to more intense strain burst
characteristics. The results demonstrate that thermal damage has some influence on strain burst
behaviour of brittle rock.
Table 5.4 Temperature influence on strain burst stress, total elastic strain energy, kinetic energy
and ejection velocity of the fragments
Temperature (°C) 25 50 75 100 125 150
Strain burst stress (%) 0 -15.7 -32.2 -44.6 -35.9 -15.8
Total elastic strain energy (%) 0 -22.9 -54.1 -68.2 -58.9 -26.9
Kinetic energy (%) 0 -22.1 -92.8 -96.3 -73.4 -27.9
Ejection velocity of the fragments (%) 0 -16.3 -70.0 -82.0 -57.2 -34.3
5.4 - Discussions
Strain burst stresses for the samples exposed to temperatures up to 100 °C declined by 44.6%,
compared to the stresses of the specimens at the room temperature (25 °C) (see Figure 5.8). It
is believed that creation of new micro-cracks due to temperature exposure led to a weakening
of the bonding among mineral grains of the samples, which can be attributed to the anisotropy
in the thermodynamic properties of different rock minerals, and this caused a degradation of
the overall rock strength. The failure mechanism for the granite specimens exposed to
temperatures up to 100 °C might have been due to intergranular fracture mechanism in which
micro-cracks first develop at the mineral grain boundaries that was consistent with the existing
literature (Yin, et al., 2012; Zuo et al. 2014; Li et al. 2016; Feng et al. 2017). As the temperature
increased from 100 up to 150 °C, the strain burst stress showed a gradual rise. It is believed
that the closure of pre-existing micro-cracks due to the thermal expansion of mineral grains by
high temperature may render the rocks denser and more compact (Funatsu et al. 2014; Gautam
et al. 2016). In order to understand this phenomenon, SEM analysis needs to be conducted,
which is a subject of our future work. However, experimental evidence in the literature suggests
that the above-mentioned mechanisms of intergranular and transgranular thermal cracking
could be behind the observed behaviour in this study. In fact Zuo et al. (2014) and Feng et al.
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conditions
(2017) reported that when the temperature was more than 100 °C, the coupled fracture
mechanism of intergranular and transgranular thermal cracking (in which the micro-cracks
develop within the mineral grains) was the main mechanism for improved compactness of the
specimens after the gradual closure of the pre-existing defects in the crystal.
Since the effects of the microcracking process are related to the magnitude of the AE events,
damage evaluation will be better understood with cumulative AE energy. It was observed that
the rate of thermal damage accumulation increased as the temperature increased from room
temperature (25° C) up to 100 °C. It is believed that the weakening of the minerals’ bonding
caused a mechanical degradation on the strength of the rocks and this triggered the rapid
thermal damage accumulation and bursting. On the other hand, when the temperature increased
from 100 °C to 150 °C, the granite specimens exhibited slower damage accumulation and
revealed intense strain burst. This can be attributed to the improved densification of the samples
due to the thermal dilation of mineral grains which decreased the distance between the
interfaces of the minerals and their mutual attraction was enhanced.
From an energy point of view, kinetic energies of the granite specimens were calculated to
assess the influence of thermal damage on the intensity of strain burst. The samples treated
with temperatures from room temperature (25 °C) to 100 °C manifested dramatically less
intense strain burst associated with slower particle ejection velocities due to the thermal
damage. At temperatures from 100 °C to 150 °C, more intense strain burst was displayed with
faster rock fragment ejection. It is believed that this increase in kinetic energy was caused by
the enhanced compactness of the samples due to the fact that thermally-induced volumetric
expansion of minerals led to the closure of the pre-existing micro-cracks and original defects
in the samples.
The aforementioned experimental results give useful enlightenments about the impact of
thermal damage on strain burst characteristics. However, more experiments considering higher
temperature levels should be performed to better understand the mechanism of strain burst
under high geo-stress and high-temperature conditions.
5.5 - Conclusion
In this chapter, temperature influence on the strain burst behaviour of granite samples was
investigated using a unique true-triaxial strain burst testing system. Based on acoustic emission,
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conditions
stress and kinetic energy analyses conducted on granite samples exposed to various
temperatures the following conclusions can be drawn:
1. The strain burst stress of granite changes with temperature from room temperature 25 °C
to 150 °C. A temperature level of 100 °C was identified as the critical transition
temperature, which induces the change in the strain burst behaviours of granite. As the
temperature increased from 25 °C to 100 °C, the strain burst stress diminished by
approximately 45%. It is believed that this declining trend is caused by the development
of microcracks that are induced by temperatures. At 100-150 °C, the strain burst stress
showed a slightly rising trend, but it is still less than that at room temperature. This can be
attributed to the improved compaction of the grains in brittle rock by the closure of pre-
existing micro-cracks due to the thermal expansion of minerals at higher temperatures.
2. The evolution of AE characteristics can be divided into three deformation stages. Those
stages are the AE quiet linear elastic deformation stage, AE growth stage and AE active
strain burst stage. The cumulative AE energy showed a sharp increase at the initial stage,
then accumulated slowly during the stress maintenance phase before increasing
dramatically until strain burst occurred. Corresponding with the failure characteristics of
the granite specimens exposed to different temperature conditions, the total cumulative AE
energy and cumulative AE counts decreased as the temperature increased from 100 °C to
150 °C. It was found that cumulative AE energy characteristics reflect the damage
evolution better as the size of micro-cracks are related to the magnitude of the AE events.
Moreover, when the temperature increased, a low-frequency band was observed due to the
thermal damage inside the specimens, which can also be an indicator for strain burst.
3. The thermal damage for strain burst (𝐷 ) increased the rate of bursting at ~95% of
𝑆𝐵
normalised axial stress levels. This can be due to the fact that as temperature caused
thermally induced micro-cracks that helped to reduce the accumulated energy at the initial
loading stage. A good relationship was observed between the trend of the b-values and the
micro- and macro- cracking during the strain burst test. The estimated b-values showed a
continuously declining trend during the test indicating that a large amount of macro-cracks
were generated prior to strain burst. Therefore, b-value analysis can be used as a precursor
to assess the degradation of the rock and strain burst process.
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Chapter 6: Conclusions and recommendations
6.1 - Introduction
The final chapter of this thesis presents the strain burst proneness indexes and criteria proposed
to evaluate the propensity of strain burst and the summary of the work done in this research,
providing conclusions and recommendations for future work. Firstly, the methodology
presented for prediction of strain burst in deep underground mines is discussed. Secondly, the
main contributions of this research are summarised. Finally, a list of recommended future work
is given followed by some additional research questions inspired by this study.
6.2 - Quantifying the influence of intrinsic rock parameters on strain burst and
application to real engineering problems
As mining progresses to greater depths, the rate and severity of strain burst hazards encountered
tend to inevitably increase, resulting in significant operational and safety challenges. Strain
burst is a sudden and violent rock fracturing and spontaneous instability phenomenon
accompanied by the abrupt release of strain energy of an excavation whereby the rock mass
rupture is initiated by mining-induced, or dynamic stress changes until the rock mass strength
(critical strain burst stress level) is reached. Such a failure characteristic poses a serious threat
to the safety and efficiency of deep underground engineering operations. Therefore, the
research on strain burst mechanism and prediction have become one of the key scientific and
technical problems in rock mechanics field.
Determination of strain burst proneness of rock is one of the challenging issues in the field of
strain burst research. Timely identification of potential precursor information enables effective
and specifically targeted measures to mitigate strain burst hazards. Is it possible to forecast
strain burst before it occurs? How can the magnitude of potential strain burst be predicted?
What magnitude of measures should be taken into account for eliminating, or minimising the
risk of strain burst and its destructive consequences to an acceptable level? These real
engineering application related questions will be explained in this chapter under strain burst
proneness assessment section. This chapter critically assesses the underlying mechanism and
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ADE | Conclusions and recommendations
consequences of strain burst evaluation methods and proposes a new energy based indexes for
practical use in real engineering applications in geomechanics.
There have been many research conducted to assess the potential risk, vulnerability and
proneness of strain burst and some discriminant indices of criterion were proposed including
the elastic strain energy storage index (Kidybinski 1981), the rock brittleness index (Wang and
Park 2001), the decrease modulus index (Singh 1989), the burst potential index (Mitri et al.
1999). Cook (1966) pointed out the significance of energy release for inducing rock burst and
proposed the energy release rate index as rockburst prediction. The burst potential index was
proposed by Mitri et al. (1999) to evaluate the potential rockburst risk after excavation and it
was stated that rockburst tends to occur when the rock energy storage rate reaches the limit of
energy storage. Kidybinski (1981) proposed the elastic strain energy index to assess the
intensity of rockburst. Wiles (2002) studied the correlation between pillar burst and the local
energy release rate and provided an indicator that can be used for predicting the potential for
rockburst. Recently, Weng et al. (2017) investigated the energy accumulation and dissipation
characteristics of rockburst failure process and they introduced a strain energy density index
for examining the energy distribution in the surrounding rock mass when rock fails due to strain
burst or spalling. Table 6.1 presents some examples of empirical criteria of strain burst
proneness in the literature which were derived from the mechanical parameters obtained by
laboratory tests.
Table 6.1 Example indices for strain burst prediction
Index or equation Explanation Reference
Ratio of the maximum tangential stress to Russenes 1974; Hoek
𝜎 /𝜎
𝜃 𝑐
the uniaxial compressive strength of rock and Brown 1980
Ratio of the elastic energy stored to the
Elastic strain energy
dissipated energy in one cycle of cycling Kidybinski 1981
index
compression test
Ratio of the energy storage rate to the
Burst potential index
maximum strain energy that the rock Mitri et al. 1999
(BPI=ESR/E)100%
mass can sustain before failure
Ratio of the compressive strength to the
Rock mass index Palmstrom 1995
tangential stress
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ADE | Conclusions and recommendations
The ratio of square of the uniaxial
Rock brittleness index compressive strength of to double amount
Wang and Park 2001
(𝑃𝐸𝑆 = 𝜎2/2𝐸 ) of the unloading tangential elastic
𝑐 𝑢
modulus
Local energy release The difference in energy stored in the
Jiang et al. 2010
rate index rock mass before and after brittle failure
Strain energy density Demonstrating the strain energy
Weng et al. 2017
index accumulation and dissipation
The ratio of the energy release of an
Rockburst energy
element generating brittle failure to the Xu et al. 2017
release rate index
limit energy storage of that element
Damage accumulation leading to strain burst is a static process followed by the dynamic release
of stored strain energy in which stored strain energy is converted to kinetic energy as in the
form of ejections of rock fragments. Therefore, strain burst from beginning to the ending is
combined quasi-static and dynamic behaviour. In this respect, to fully understand the strain
burst mechanism it is essential to consider quasi-static and dynamic parameters for forecasting
the potential and intensity of strain burst. Although the strength and deformability of rocks can
be approximately predicted, the intrinsic structure and the internal failure mechanism still
remain for further investigation. Due to the complex physical and mechanical properties of
rock mass, the main causes related geomechanical properties and the strain burst mechanism
present a challenging concern to researchers in rock mechanics.
6.3 - New strain burst proneness indexes based on excess stored strain energy
In this section, strain burst characteristics based on the energy theory was analysed and energy
indexes were proposed to quantitatively evaluate the intensity of strain burst of brittle rock.
Based on the energy evolution characteristics of brittle granite under uniaxial and triaxial
compression, true-triaxial loading-unloading and three-point bending, new strain burst
proneness indexes were proposed and new strain burst criterion based on these indexes were
presented. Note that these indexes were proposed for brittle hard granite.
6.3.1 - The excess strain energy index 𝛀
𝑺𝑩
According to the circumferential-strain controlled uniaxial and triaxial compression tests, the
elastic stored strain energy, fracture energy and excess strain energy that is the potential energy
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ADE | Conclusions and recommendations
for strain burst, of the granite specimens during the entire loading were accurately calculated
and the rule of energy accumulation and release in granite was systematically analysed. It was
found that the maximum strain energy stored and excess strain energy in the rock are affected
by the confining pressure and temperature.
Based on the above-mentioned theory, here a new energy index for strain burst proneness Ω
SB
was proposed, can be calculated as in Equation 6.1:
dΦ
EX
Ω = (6.1)
SB dU
E
where dΦ and dU are the excess strain energy released during brittle failure (strain burst)
EX E
and the elastic stored strain energy after Class II behaviour starts, respectively. The energy
calculations are shown as follows (see Chapter 3):
𝜎2
𝑑𝑈 = 𝐴 (6.2)
𝐸 2𝐸
𝜎𝐵 𝜎2 −𝜎2 (𝑀−𝐸)
dΦ = ∑ 𝑖 𝑖+1 (6.3)
𝐶𝑊 2𝐸𝑀
𝑖=𝜎𝐴
𝜎𝐶 𝜎2 −𝜎2 (𝑀−𝐸)
dΦ = ∑ 𝑖 𝑖+1 (6.4)
𝐹𝑀 2𝐸𝑀
𝑖=𝜎𝐵
𝜎2
𝑑𝑈 = 𝐶 (6.5)
𝑅𝐸 2𝐸
dΦ = 𝑑𝑈 −dΦ −dΦ −𝑑𝑈 (6.6)
𝐸𝑋 𝐸 𝐶𝑊 𝐹𝑀 𝑅𝐸
where Φ is the energy consumption dominated by cohesion degradation during stable
𝐶𝑊
fracturing, Φ is the energy dissipated during the mobilisation of frictional failure, 𝑈 is the
𝐹𝑀 𝑅𝐸
residual stored elastic strain energy, 𝜎 is the point of axial strain reversal, 𝜎 is the point of
𝐴 𝐵
brittle failure intersection (see Figure 3.4 in Chapter 3), 𝐸 is the elastic stiffness of the specimen
and 𝑀 (𝑀 = 𝛿𝜎/𝛿𝜀) is the post-peak modulus between two incremental stress points, 𝜎 and
𝑖
𝜎 which can vary significantly with the fracture development.
𝑖+1
From the above analyses, the strain burst proneness of the thermally-treated granite specimens
at different confining pressure can be classified into three grades: low, medium and strong
strain burst proneness. The grading standards of strain burst proneness based on Ω are listed
SB
in Table 6.1. According to the calculated Ω and the failure pattern of the granite specimens
SB
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(see Figure 3.8 in Chapter 3), a new criterion for strain burst proneness with Ω was proposed
SB
as follows:
Ω > 0.08 low strain burst proneness
SB
0.04 < Ω < 0.08 medium strain burst proneness
Confinement SB (6.7)
Ω < 0.04 strong strain burst proneness
SB
Ω < 0.2 low strain burst proneness
SB
0.2 < Ω < 0.4 medium strain burst proneness
Temperature SB (6.8)
Ω > 0.4 strong strain burst proneness
SB
Table 6.2 Classification of strain burst proneness using the excess strain energy index Ω
SB
Confining pressure (MPa) 𝛀 Strain burst proneness
𝐒𝐁
0 0.187 Low
0 0.272 Low
0 0.205 Low
10 0.071 Medium
20 0.079 Medium
20 0.087 Medium
30 0.021 Strong
30 0.040 Strong
40 0.037 Strong
40 0.038 Strong
50 0.024 Strong
60 0.007 Strong
Figure 6.1 presents the influence of confining pressure and temperature on strain burst
proneness. It can be seen that the strain burst proneness of brittle granite is strongly dependent
on the pre-heating temperature and confinement. The results demonstrated that the higher the
confining pressure and temperature, the stronger the strain burst proneness will be. It is
believed that due to the anisotropy in the thermodynamic properties of different rock minerals,
the amount and width of the microcracks inside the specimen increased, and this triggered the
rapid thermal damage accumulation and bursting. In other words, the fundamental reason for
the increase of strain burst proneness is the thermally induced damage by microcracking.
Thermally induced damage caused less elastic strain energy accumulation and hence the excess
strain energy which is a measure for the intensity of the intrinsic strain burst in the rock
decreased with increasing temperature, resulting in stronger strain burst proneness.
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6.3.2 - Released energy index 𝛌
𝑺𝑩
The kinetic energy of the ejected fragments during strain burst can serve as a significant
precursor for evaluating the strain burst intensity quantitatively. Using a high-speed camera,
the ejection failure process of rock fragments were observed in true-triaxial loading-unloading
strain burst tests. The ejection velocities and kinetic energies from the tested granite specimens
were quantitatively estimated by analysing the recorded videos. After measuring the velocity,
the total kinetic energy of the ejected fragments was calculated by using Equation 6.9.
𝑛
1
2
𝐸 = ∑ 𝑚 𝑣 (6.9)
𝑘 2 𝑖 𝑖
𝑖=1
where n is the number of fragments having D > 10 mm and m > 0.5 g, 𝑚 is the mass of the 𝑖th
𝑖
rock fragment and 𝑣̅ is the initial ejection velocity of the 𝑖th rock fragment. By using the
𝑖
equation above, the total kinetic energies for all granite specimens treated with different
temperatures were calculated. In addition, the strain burst stress (𝜎 ) and total elastic strain
𝑆𝐵
energy (U ) of the granite samples exposed to different temperatures were calculated.
E
Based on the kinetic energy and stress analyses, a released energy index 𝜆 for strain burst
SB
proneness, was proposed, which can be described by:
𝐸
𝑘
𝜆 = (6.10)
SB U
E
According to the calculated 𝜆 of the thermally-treated granite specimens, a new criterion for
SB
strain burst proneness with index 𝜆 was proposed as follows:
SB
𝜆 < 0.5 low to moderate strain burst
SB
(6.11)
λ > 0.5 medium to intense strain burst
SB
𝜎𝑆𝐵
> 1.2 low strain burst proneness
𝜎𝑈𝐶𝑆
1 <
𝜎𝑆𝐵
< 1.2 moderate strain burst proneness (6.12)
𝜎𝑈𝐶𝑆
𝜎𝑆𝐵
< 1intense strain burst proneness
𝜎𝑈𝐶𝑆
where 𝜎 is the uniaxial compressive strength.
𝑈𝑆𝐶
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ADE | Conclusions and recommendations
The strain burst proneness of thermally-induced granite specimens is given in Table 6.2. It can
be seen that strain burst proneness increased with an increased temperature which can be
attributed to the mechanical strength degradation induced by thermal microstructures,
rendering the rock relatively weaker.
Table 6.3 Classification of strain burst proneness using the released energy index 𝜆 and
𝜎𝑆𝐵
SB
𝜎𝑐𝑚
𝝈
𝑺𝑩
Temperature (°C) 𝛌 Strain burst proneness
𝑺𝑩 𝝈
𝒄𝒎
0.586 1.67 Low
25
0.409 1.50 Low
0.572 1.24 Low
50
0.322 1.87 Low
0.056 1.02 Moderate
75
0.099 1.13 Moderate
0.058 1.06 Moderate
100 0.059 0.92 Intense
0.065 0.65 Intense
0.174 1.11 Moderate
125 0.439 0.98 Intense
0.419 0.96 Intense
0.554 1.06 Moderate
150
0.439 1.67 Low
6.3.3 -Energy release rate index 𝚿
𝑺𝑩
Energy release rate which is a measure of the energy that is dissipated per unit increase in an
area during crack growth is important for the successful assessment of fracturing characteristics
during strain burst. In this study, the effects of various loading rates on the strain burst
proneness for thermally-treated granite was analysed and discussed. The applied energy is
equal to the work done on the crack surface for its propagation which can be determined by the
applied load and the displacement in the system.
Based on the above-mentioned theory, an energy release rate index Ψ for strain burst
SB
proneness was presented, as follows:
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ADE | Conclusions and recommendations
𝐺
Ψ = 𝐼 (6.13)
SB 𝑊
Where 𝐺 and 𝑊 are the energy-release rate and applied energy on granite under mode I
𝐼
fracture, respectively.
A strain burst proneness criterion based on Ψ index was proposed (see Equation 6.14) and
SB
the coupled influence of loading rate and temperature on strain burst proneness was
investigated in this study.
Ψ > 1 (Low strain burst proneness)
SB
0.75 < Ψ < 1 (Moderate strain burst proneness) (6.14)
SB
Ψ < 1 (Intense strain burst proneness)
SB
The detailed strain burst proneness of granite under various levels of temperature at different
loading rates are given in Table 6.3. The results showed that the strain burst proneness
decreases with increasing loading rate as the strength and fracture toughness of granite,
resulting in slight strain burst proneness. Increased temperature, on the other hand, caused
stronger strain burst proneness of granite due to the thermal damage resulting in deterioration
of the tensile stress resistance.
Table 6.4 Classification of strain burst proneness using the energy release rate index Ψ
SB
Temperature (°C) Loading rate (mm/min) 𝚿 Strain burst proneness
𝑺𝑩
0.02 0.641 Intense
0.05 1.242 Low
RT
0.08 1.265 Low
0.1 1.484 Low
0.02 0.737 Intense
0.05 1.045 Low
100
0.08 0.837 Moderate
0.1 1.144 Low
0.02 0.631 Intense
0.05 0.847 Moderate
175
0.08 0.479 Intense
0.1 0.731 Intense
250 0.02 1.016 Low
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0.05 0.752 Intense
0.08 0.947 Moderate
0.1 0.840 Moderate
Based on the above-mentioned strain burst proneness indexes and criteria, a methodology for
forecasting the propensity of strain burst is proposed, as depicted in Figure 6.2. Using these
indexes can provide guidelines for the development of an effective and reliable method to
forecast the propensity of strain burst. According to the energy calculations in this new testing
methodology, calculating the excess strain energy, stored elastic strain energy and energy
release rate evolution characteristics can be used for improved understanding of the
performance and design of rock support systems in strain burst-prone mines. Appropriate rock
support design can be provided by considering the energy absorption capacity of rock support
and the energy characteristics obtained from the laboratory tests conducted for investigating
the underlying mechanism of strain burst damage. Therefore, this research will lead to better
and more efficient prediction methods for brittle rock failure and strain burst, towards planning
guidelines and ultimately safer deep underground working environments.
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6.4 - Conclusions
The objective of this research is fourfold: first, to investigate the energy evolution
characteristics during strain burst by conducting circumferential strain controlled tests under
the combined influence of thermal damage and confining pressure, and second determining
quasi-static and dynamic fracture toughness on thermally treated Australian CCNSCB granite
specimens at various loading rates and examine the relation between the quasi-static and
dynamic mode-I fracture toughness and energy release rates; third, investigate the influence of
deviatoric stresses and temperature effects on strain burst behaviour using rectangular prism
granite specimens exposed to different pre-heating temperatures under true-triaxial
loading/unloading conditions; and finally proposing strain burst criteria or index for strain burst
proneness by the results from the tests mentioned above and upscale these finding to apply for
the real engineering applications.
Apart from this, three other motivating branches of interest can be directed to systematically
and thoroughly assess the influence of external factors including confining pressure, thermal
damage and loading rate on the mechanical properties and energy characteristics of brittle
Australian granite during strain burst in deep mining operations.
Based on the acoustic emission, stress, kinetic energy analyses and fracture characterisation
carried out on granite samples exposed to various temperature, confinement and loading rate
the following key conclusions can be drawn:
Forecasting the propensity of strain burst
1. An energy calculation method was developed based on post-peak energy analysis. AE
responses during compression tests were used to assess the energy and crack evolution
characteristics of Australian granite specimens under different confinement. Using AE
characteristics, fracture energy was split into two-class: 1) energy consumed dominantly
by gradual weakening of cohesive behaviour and 2) energy dissipated during the
mobilisation of frictional failure. A portion of elastic energy, released from the Class II
rock, was defined as excess strain energy which is a measure for the propensity of the
intrinsic strain burst in the rock. It directly determines the intrinsic ejection velocity of the
rock fragments when a bursting event occurs. Therefore, this methodology can be used for
quantitative predictions of bursting strain energy in the field which could facilitate
174 |
ADE | Conclusions and recommendations
improving the early warning efficiency and provides a comprehensive guideline for the
mitigation methods to reduce strain burst intensity.
2. Confinement has significantly affected the post-peak energy redistribution characteristics
and fracture mechanism of granite. The elastic stored strain energy, energy consumed by
dominating cohesion weakening, and energy dissipated during mobilisation of frictional
failure were 8.74, 2.53 and 12.1 times the values at unconfined condition, resulting in more
severe strain burst indicating that rising up the confining pressure improved the efficiency
of energy accumulation. This explains why the damage degree of granite is more
prominent in the process of deep excavations.
3. The temperature has significantly affected the post-peak energy redistribution
characteristics and fracture mechanism of granite. The elastic stored strain energy, total
fracture energy, excess strain energy diminished by 80, 82 and 43%, respectively when the
temperature increased from room temperature to 250 °C. This declining trend was
attributed to the development of micro-cracks that were induced by elevated temperatures.
Thermally induced damage caused less strain energy accumulation and hence the excess
strain energy decreased with increasing temperature. Another parameter to express the
intensity of a burst event, ejection velocity, dropped down as the gradual increase of
temperature. The proposed approach can provide an early warning of brittle rock
instability, which is significant for strain burst assessment in deep mining operations.
4. The fracturing mechanism of granite was influenced by both confining pressure
(excavation depth) and temperature. The dominant failure pattern of granite changed from
multiple splitting failure to splitting-shear composite failure as the level of confinement
increased. When the temperature was less than 100 °C, granite samples experienced more
induced intergranular thermal fracturing. Coupled fracture mechanism of intergranular and
transgranular thermally induced cracking were the main fracture mechanism triggering
strain burst when the temperature exceeded 100 °C.
175 |
ADE | Conclusions and recommendations
Quasi-static and dynamic fracture characterisation
1. The CCNSCB specimen combines the merits of two ISRM-suggested methods (CCNBD
and NSCB methods), and thus it allows accurate determination of the mode I fracture
toughness of granite under quasi-static and dynamic loadings.
2. The experimental results indicated that the quasi-static fracture toughness and energy-
release rate in mode I are a function of loading rate and they presented a rising trend with
increasing loading rate. At high loading rates, transgranular fractures became dominant
which consumed more energy than intergranular fractures; this in turn, resulted in more
straight fracture path and posed a less rough fracture surface when compared to the low
loading rate condition.
3. Under the same loading rate, the quasi-static mode I fracture toughness and energy-release
rate of granite showed a gradual fall (17% and 30%, respectively) with ascending
temperature from 25 °C to 250 °C due to the thermally-induced micro-cracks within the
rocks. These findings of this investigation will be useful in achieving a better
understanding of initiation of fracturing during strain burst under various temperature and
loading rate conditions.
4. The stress-strain curves of granite under various impact velocities and temperatures
showed the same deformation stages; elastic deformation, yielding and failure. When the
impact velocity was high, the loading rate strengthening effect became more remarkable
and the strength of granite increased under all temperatures. The failure modes of
Australian granite also exhibited rate dependence at the same temperature level. Along
with the high impact velocity, the failure mode of the pre-heated granite changed from
tensile splitting (characterisation of Class I) to pulverisation or breaking into many small
pieces in which the specimens were pulverised by the excess energy in Class II loading.
Under the same dynamic impact, an increase in the treatment temperature weakened the
interaction force between the particles and aggravated the fragmentation degree of granite.
5. The DIFT of Australian granite was obtained by the quasi-static analysis that was
evidenced by the dynamic force balance until the time to fracture. The DIFT of the granite
presented an ascending trend with the loading rate at a given heat-treatment temperature
176 |
ADE | Conclusions and recommendations
and decreased with increasing temperature, revealing the deterioration of the ability to
resist fracturing with the rise of temperature. Therefore, in order to effectively crush the
deep rock, a favourable measure should be applied to reduce the intensity of strain burst
by considering a combined application of a thermal treatment and impact with a proper
loading rate.
Effects of thermal damage on strain burst mechanism for brittle rocks under true-
triaxial loading-unloading conditions
1. The strain burst stress of granite changes with temperature from room temperature 25 °C
to 150 °C. A temperature level of 100 °C was identified as the critical transition
temperature, which induces the change in the strain burst behaviours of granite. As the
temperature increased from 25 °C to 100 °C, the strain burst stress diminished by
approximately 45%. It is believed that this declining trend is caused by the development
of microcracks that are induced by temperatures. At 100-150 °C, the strain burst stress
showed a slightly rising trend, but it is still less than that at room temperature. This can be
attributed to the improved compaction of the grains in brittle rock by the closure of pre-
existing micro-cracks due to the thermal expansion of minerals at higher temperatures.
2. The evolution of AE characteristics can be divided into three deformation stages. Those
stages are the AE quiet linear elastic deformation stage, AE growth stage and AE active
strain burst stage. The cumulative AE energy showed a sharp increase at the initial stage,
then accumulated slowly during the stress maintenance phase before increasing
dramatically until strain burst occurred. Corresponding with the failure characteristics of
the granite specimens exposed to different temperature conditions, the total cumulative AE
energy and cumulative AE counts decreased as the temperature increased from 100 °C to
150 °C. It was found that cumulative AE energy characteristics reflect the damage
evolution better as the size of micro-cracks are related to the magnitude of the AE events.
Moreover, when the temperature increased, a low-frequency band was observed due to the
thermal damage inside the specimens, which can also be an indicator for strain burst.
3. The thermal damage for strain burst (𝐷 ) increased the rate of bursting at ~95% of
𝑆𝐵
normalised axial stress levels. This can be due to the fact that as temperature caused
177 |
ADE | Conclusions and recommendations
thermally induced micro-cracks that helped to reduce the accumulated energy at the initial
loading stage. A good relationship was observed between the trend of the b-values and the
micro- and macro- cracking during the strain burst test. The estimated b-values showed a
continuously declining trend during the test indicating that a large amount of macro-cracks
were generated prior to strain burst. Therefore, b-value analysis can be used as a precursor
to assess the degradation of the rock and strain burst process.
4. The kinetic energy of the ejected fragments dramatically decreased until they reached the
critical temperature of 100 °C. This is because of manifested thermally induced damage
which caused less elastic strain energy accumulation. When the temperature increased
from 100 °C to 150 °C, kinetic energy had also a slight rise which is associated with the
higher initial velocity of ejected fragments which may occur due to the expansion of
mineral grains by increased temperature. This helped to improve the compactness of the
rock which implies that a more intense or severe strain burst may be encountered in
situations where temperatures rise above the critical temperature of 100 °C.
Quantifying the influence of intrinsic rock parameters on strain burst and
application to real engineering problems
1. To estimate and classify the strain burst proneness of brittle rock, energy evolution
characteristics of granite were used to assess the tendency of strain burst. Excess strain
energy (Ω ), released energy (𝜆 ), and energy-release rate (Ψ ) indexes were proposed
SB SB SB
on the basis of energy characteristics for brittle rock.
2. Based on the strain burst proneness of granite specimens obtained through circumferential-
strain controlled uniaxial and triaxial compression tests, true triaxial loading-unloading
strain burst tests, and three-point bending mode I fracture toughness tests, and the indexes
proposed, new criterions for strain burst proneness were put forward. The influence of
confining pressure, temperature and loading rate on the strain burst proneness was also
analysed and discussed.
178 |
ADE | Conclusions and recommendations
6.5 - Recommendations for future work
In addition to the results reported in this thesis, the following interests can be recommended
for future work:
1. Conducting circumferential-strain controlled tests with simultaneously increasing the
temperature and confining pressure.
2. The growth of the microcracks in rocks is accompanied by significant inelastic
deformation near the crack tip. This highly damaged region adjacent to the crack tip is
called a fracture process zone (FPZ) within the material undergoes micro-damaging. In the
FPZ, micro-cracks close or open depending on their orientation with respect to the
direction of the applied load, and crack growth, in fact, occurs by connecting the micro-
cracks at a critical load. Therefore, PFZ during strain burst should be analysed and
discussed more in-depth to estimate the PFZ in underground excavation and thus more
appropriate supporting system can be applied with some economic benefits.
3. In the view of the study on dynamic fracture properties of rock under coupling of
temperature and static pressure will to be carried out for a better understanding of dynamic
fracture characteristics during strain burst.
4. 3D X-ray micro-CT technique deserves examination for accurately quantification of the
thermally-induced damage under different loading conditions.
5. The effects of confining pressure on the dynamic fracture parameters of brittle rock should
be studied to understand the fracture propagation characteristics under confined
environment. This will help to identify the initiation of unstable crack growth.
6. The effect of intermediate principal stress on rock failure is commonly acknowledged, and
it was first verified that, under constant 𝜎 condition, the rock strength in the conventional
3
triaxial extension was higher than that in the conventional triaxial compression test.
Therefore, the influence of intermediate stress on strain burst mechanism under true-
triaxial unloading conditions should be subjected to detailed investigation.
7. The influence of loading and unloading rate on strain burst behaviour under true-triaxial
loading-unloading conditions should be studied.
8. The energy dissipation due to the formation of rock fragments triggered by tension and
shear failures during strain burst process should be systematically investigated.
179 |
ADE | Abstract
Hydraulic simulation models have been used to simulate the steady-state of a water
distribution system (WDS) for serval decades. These models have been used in WDS
simulationtoolkitsandhaveplayedacriticalroleinthedesign,operation,andmanagement
of WDSs in industry and research. In recent years, a number of graph theory based
WDSsolutionmethodshavebeendeveloped. Thesemethodshaveexploredthestructural
properties (both matrix and graph) of the problem to improve the speed and reliability of
WDSsimulations. Onequestionthatnaturallyarisesis whichmethodor combinationof
methodsshouldbeapplied?
In this thesis, a WDS simulation testbed, called WDSLib, has been developed as a
toolthatcanbeusedtoanswertheabovequestion. WDSLibisanextensiblesimulation
toolkit for the steady-state analysis of a WDS. It has been created using modularised
object-oriented design and implemented in C++ programming language. WDSLib can
be used (1) to implement, test, and compare different solution methods, (2) to focus the
researchonthemosttime-consumingpartsofasolutionmethod,(3)toguidethechoice
of solution method when multiple simulation runs are used (such as occurs in a genetic
algorithmrun).
WDSLibhasbeenusedtoinvestigatetheperformanceoffoursolutionmethods,namely
the global gradient algorithm (GGA), the reformulated co-tree flows method, the GGA
withtheforest-corepartitioningalgorithm(FCPA),andtheRCTMwiththeFCPA,oneight
casestudybenchmarknetworkswithbetween934and19647pipesandbetween848and
17971 nodes. The results can be used to inform the choice of the solution method for a
givencombinationofthenetworkfeaturesunderdifferentdesignsettings. Thisworkalso
demonstrateshowto(1)usetheWDSLibtoimplement,test,andbenchmarktheexisting
solution methods and (2) use the results to determine which method or combination of
methodstousedunderasettingofinterest.
Anewgraphtheoryalgorithm,calledthebridge-blockpartitioningalgorithm(BBPA),
has been proposed which further partitions the WDS network in a number of bridge
components and a number of block components. The BBPA is also implemented in the
WDSLib in order to ensure a fair comparison with the existing methods. The BBPA
is a pre-processing and post-processing method, the use of which provides significant
advantages over the current methods in terms of both the computational speed and the
reliability of the solution. This work also demonstrates how to (1) use the WDSLib to
implement, test, and benchmark the new solution method and (2) use the WDSLib to
demonstrate the efficiency of new method without having to reengineer the content of
sharedWDSLibfunctionsanddatarepresentations.
iii |
ADE | Chapter1. IntroductionandPublicationsOverview
In a hydraulic simulation, there are two sets of primary equations that govern the
underlyingrelationshipsofaWDSundersteady-stateconditions: asetofmassconservation
or continuity equations and a set of energy conservation equations. Some assumptions
aremade tosimplify thegoverningequations ofa hydraulic simulationincluding: (1)that
the velocity heads are negligible when compared to the friction head losses, (2) that the
minorheadlosses atthepipejunctionsandfittings aremuchsmallerthanthefrictionhead
losses, (3)water isincompressible, (4)thedemandsare consideredtooccur ataparticular
time instance and are concentrated at the nodes of a network, and (5) the demands are
independentof nodalpressure. Withthe aboveassumptions,the twogoverningequations
mentionedabovecanbedescribedas: (1)themassconservationequations: thetotalinflow
mustequaltothetotaloutflowatanynode;(2)theenergyconservationequations: thehead
differencemustbeequaltothefrictionheadlossforanypipe.
Thesetwosetsofgoverningequationscanbeformulatedasalargeandsparsenon-linear
saddle point problem (Benzi et al. 2005). There is a number of well-known iteration
methods for solving this non-linear saddle point problem. These include: range space
methods(TodiniandPilati1988), nullspacemethods (Rahal 1995;Elhayetal. 2014),and
loop-based methods (Epp and Fowler 1970; Nielsen 1989). Moreover, the use of graph
theoryhasbecameincreasinglypopularindevelopingsolutionmethodstoimproveboth
the efficiencyand thereliability of WDSsolution process. Themain reasonunderpinning
thephilosophyofusinggraphtheorywithhydraulicsimulationistheinvariantnatureofthe
networktopology. Thisfixedtopologycanoftenbeexploitedasapre-and-post-processing
steptospeed-upthecomputations.
Range Space Methods: Theglobalgradientalgorithm(GGA)(TodiniandPilati1988),
a range space method, employed block elimination to reduce the size of the key matrix.
Although graph theory is not used when deriving GGA solution method, the node-arc
incidencematrix, whichwasfirstusedin Todiniand Pilati(1988)todescribe thenetwork
topology, provides a portal into using graph theory to simplify the solution process of a
WDSnetwork. Simpsonetal.(2012)developedtheconceptofseparatingtheforestand
core components while Deuerlein (2008) introduced the forest-core partitioning algorithm
(FCPA). The forest component is separated out from the core by sweeping the node-arc
incidence matrix. After the forest component is separated out, a standard GGA is then
applied to the core component of the network. The main advantage of the FCPA is to
separatetheforest,whichislinearcomponentofthesystemofequations,fromthecore,
whichisthenonlinearcomponentofthesystemofequations. Thisprocessspeedsupthe
demand-dependentmodel(DDM)solutionprocesswhenanetworkhasasignificantforest
portion. Later, the graph matrix partitioning algorithm (GMPA) (Deuerlein et al. 2015)
wasproposed. TheGMPAexploitedthelinearrelationshipsbetweenflowsoftheinternal
treeswithinthecoreandtheflowsofthecorrespondingsuper-linksaftertheforestofthe
networkhadbeenremoved.
Loop-Based Methods: The Hardy Cross method (Cross 1936), a loop based method,
is the oldest method. In the Hardy Cross method, the system of equations is solved by
successiveapproximation,inwhichasetofflowsthatsatisfiescontinuityissuccessively
correctedloopbyloopuntilthepredefinedstoppingtesthasbeenmet. Inanotherpaper,
Epp and Fowler (1970) developed a programmable version of the Hardy Cross method.
However,theloop-basedmethodisnotwidelyusedbecause(1)itrequiredtheidentification
oftheloops,(2)itrequiredtheuseofapseudo-sourceifthenetworkhasmorethanone
source,and(3)itrequiredthedeterminationasetofinitialflowsthatsatisfiescontinuity.
Deuerlein(2008) proposeda decomposition model forWDS graph, inwhich thenetwork
2 |
ADE | Chapter1. IntroductionandPublicationsOverview
solution methods. Moreover, a number of graph theory based WDSsolution methods have
been efficiently implemented to provide a fast simulation platform for both once-off and
multi-runsimulationsettings.
Aim#2: Toprovideinsightinthechoiceofsolutionmethodsforgivencombinations
ofnetworkfeaturesand givendesignsettings Itisoftendifficult,ifnotimpossible,to
determine a priori what method or combination of methods to use for a given network
topology. ThesimulationplatformdevelopedinAim#1isusedtobenchmarkthehydraulic
solutionofa numberofcasestudywater distributionnetworkswithavariety oftopology
features. Thecorrelationsbetweenthesetopologyfeaturesandtherelativeperformanceof
themethodsofinterestarestudied.
Aim #3: To develop a new graph theory based algorithm to further partition the
WDS Anewalgorithmthatcanbeusedtofurtherpartitionthenetworkisproposed. This
algorithmis implementedin thesimulation platformdeveloped inAim #1anda detailed
casestudyiscarriedoutexploringthealgorithm’sefficiencyanditsreliability.
1.3 Publications
Thisthesisiscomprisedofthreepublications. Theircontributiontothebodyofknowledge
isalignedwiththeresearchaimsinSection1.2. Thissectiongivesabriefdescriptionfor
eachpublicationanditscontribution.
Chapter 3 presents the development of an extensible simulation platform, WDSLib,
for the demand-driven steady-state analysis of aWDS. WDSLib has been created using a
modularised object-orienteddesign and implementedin the C++ programming language,
and has been validated against a reference MATLAB implementation. Two solution
methods, namely the global gradient algorithm (GGA) and the reformulated co-tree
flowsmethod(RCTM),andapre-processingandpost-processingmethod,theforest-core
partitioningalgorithm(FCPA),arecurrentlyimplementedinWDSLib.
Chapter4presentsathoroughbenchmarkstudytocomparetheperformanceofGGA,
GGA with FCPA, RCTM, and RCTM with FCPA using WDSLib developed in the first
publication. Theresultsofthis studywillhelpinformthechoiceof solutionmethodsfor
givencombinationsofnetworkfeaturesandgivendesignsettings.
Chapter5proposesabridge-blockpartitioningalgorithm(BBPA)thatfurtherpartitions
the network into bridges, blocks and cut-vertices. It has been shown that the use of
the BBPA is not only able to significantly reduce the computation time of the once-off
simulation and the multi-run simulation, but also able to improve the reliability of the
solution.
1.3.1 Contributions to the development of a WDS Simulation
Platform for WDS Simulation and Optimisation
A number of contributions have been made in developing a framework for efficiently
incorporating graph theory in a WDS simulation model that can be used for simulation,
optimisation,andmanagementofaWDSnetwork. Thesecontributionsarepresentedwhile
describingtheworkflowsinvolvedindifferentgraphtheorybasedWDSsolutionmethods.
4 |
ADE | Chapter 2
Review of the Existing Water Distribution
System Solution Methods
Thischapterreviewsthefundamentalaspectsofthehydraulicanalysisofasteady-state
demand-driven water distribution system. The system of equations for a WDS is first
described in Section 2.1. Section 2.2 describes some of the recent applications of graph
theoryconceptsinthesolutionofthesteady-stateproblemforawaterdistributionsystem.
Then, inSection 2.3,the solutionmethods thatare usedto simulatethe steady-stateof a
WDSarereviewed.
2.1 WDS Model equations
Thisthesisconsidersademand-drivenwaterdistributionsystemwithn pipes,n unknown-
p j
head nodes and n fixed-head nodes. The j-th pipe of the network can be characterised
f
by its diameter d , length l , resistance factor r . The i-th node of the network can be
j j j
characterisedbyitsnodaldemandd ,andtheelevationheadz .
i i
Let q = (q ,q ,....q )T denote the vector of unknown flows, h = (h ,h ,....h )T
1 2 np 1 2 nj
denote the vector of unknown heads, r = (r ,r ,....r )T denote the vector of pipe
1 2 np
resistance factors, d = (d ,d ,.....d )T denote the vector of nodal demands, e =
1 2 nj l
(e ,e ....e )T denotethevectoroffixedheadelevations.
l
1
l
2
lnr
The head loss exponent n is assumed to be dependent only on the head loss model:
n = 2 for the Darcy-Weisbach head loss model and n = 1.852 for Hazen-Williams head
loss model. The head loss within the pipe j, which connects the node i and the node
k, is modelled by h h = r q q n 1. Denote by G(q) Rnp np, a diagonal square
i k j j j − ×
− | | ∈
matrix with elements [G] = r q n 1 for j = 1,2,....n . Denote by F(q) Rnp np,
jj j j − p ×
| | ∈
a diagonal square matrix where the j-th element on its diagonal [F] = ∂ [G] q .
jj ∂qj jj j
The unknown-head node-arc incidence matrix A is full rank, where [A ] is used to
1 1 ij
representtherelationshipbetweenpipeiandnodej: [A ] = 1ifpipeientersnodej,
1 ij
−
[A ] = 1ifpipeileavesnodej,and[A ] = 0ifpipeiisnotconnectedtonodej. The
1 ij 1 ij
matrixA isthe fixed-headnode-arc incidencematrix,where [A ] isused torepresent
2 2 ij
therelationshipbetweenpipeiandfixedheadnodej: [A ] = 1ifpipeientersfixed
2 ij
−
headnodej,[A ] = 1ifpipeileavesfixedheadnodej,and[A ] = 0ifpipeiisnot
2 ij 2 ij
connected to fixed head node j. The steady-state flows and heads in the WDS system
9 |
ADE | Chapter2. ReviewoftheExistingWaterDistributionSystemSolutionMethods
SpanningTree Aspanningtreeisanacyclicsubgraphwhichtraverseseverynodeina
graph,suchthattheadditionofanyco-treeelementcreatesaloop. Anacyclicgraphisa
graph having no graph cycles. A WDS, with or without a forest, can be partitioned into
two subgraphs: a spanning tree component, G = (V ,E ), and a set of co-tree edges,
st st st
E , so that E E = E , E E = . This relationship can sometimes be used to
ct st ct c st ct
∪ ∩ ∅
furtherexploittheblockstructureoftheJacobianmatrixtoproduce,inrealisticWDSs,an
evensmallerkeymatrix. Thisisachievedbydealingseparatelywiththespanningtreeand
theco-treeintheNewtonmethodlinearisation.
Loop A loop, know as a simple cycle in graph theory, is a path of edges and vertices
wherein a vertex is reachable from itself with no repetitions of vertices and edges. Two
loops,C andC ,canbeusedtoformanotherloopbyusingthesymmetricdifferenceof
1 2
twosets((C C ) (C C )). Thesetofallloopsiscalledthecyclespace. Consider
1 2 1 2
∪ − ∩
aconnectedgraphG=(V,E)withaspanningtreeG Gandthecomplementaryco-tree
st
∈
edges E . For every co-tree edge e E there is a unique cycle C in G +e; these
ct ct e st
∈
cyclesC arethefundamentalcyclesofGwithrespecttothespanningtreeG .
e st
IfT isaspanningtreeorspanningforestofagivengraphG,andeisanedgethatdoes
notbelongtoT,thenthefundamentalcycleC definedbyeisthesimplecycleconsisting
e
ofetogetherwiththepathinT connectingtheendpointsofe. Thereareexactlyn n +c
p j
−
fundamentalcycles,oneforeachedgethatdoesnotbelongto T . Each ofthemislinearly
independentfromtheremainingcycles,becausetheyincludeanedgeethatisnotpresent
inanyotherfundamentalcycle. Therefore,thefundamentalcyclesformacyclebasisfor
thecyclespace. Acyclebasisofagraphisaminimalsetofsimplecyclesthatallowsevery
cycleinthecyclespacetobeexpressedasasymmetricdifferenceofbasiscycles.
Minimum cycle basis The cycles that can be made by a spanning tree and the
corresponding co-tree is a subset of the cycle space. In cycle-based methods, it is often
preferable to use a shortest cycle basis. The Shortest Maximal Cycle Basis (SMCB) is
a cycle basis B of a given graph G with the property that the length of the longest cycle
includedinBisthesmallestamongallbasesofG.Itispossibletominimisethenumberof
non-zerosinthekeymatrixofloop-basedmethodsbyusingashortestcyclebasis.
2.3 Solution Methods
We consider three types of hydraulic solution methods: (1) range space methods, (2)
loop-based methods and (3) null space methods. These three types of solution methods
and the applications of graph theory in each of the three categories are discussed in the
followingsections.
2.3.1 Range Space Methods
Theglobalgradientalgorithm(GGA),arangespacemethod, was first proposedbyTodini
andPilati(1988). TheyappliedblockeliminationtoEq. (2.5)toyieldatwo-stepNewton
solverforthecaseswhentheheadlossismodelledbytheHazen-Williamformula:
h(m+1) = U 1 nd+A T [(1 n)q(k) G 1A e ] (2.6)
− 1 − 2 l
− − −
n o
11 |
ADE | Chapter3. Publication1: WDSLib: AWaterDistributionSystemSimulationTestBed
data acquisition (SCADA) operational setting, and (4) to adjust control devices, such as
valves, in a management setting. In the design setting and both the above operational
settings,repeatedhydraulicassessmentisrequiredonanetworkwithfixedtopology. Inthe
management setting, repeated hydraulic assessment is required on a network with flexible
networkparametersettings. Withever-increasingnetworksizesandtheneed forreal-time
managementusingaSCADAsystem,itisimportanttohavearobustsimulationpackage
whichcanbeconfiguredtobemaximallyefficientwhateverthesetting.
Inthefieldofhydraulicsimulation,thesystemofequationscanbeformulatedasalarge
andsparsenon-linearsaddlepointproblem. Thereareseveralwell-knowniterationmethods
for solving the non-linear saddle point problem. These include: range space methods
(GlobalGradientAlgorithm(TodiniandPilati1988)),Nullspacemethods(Co-Treeflow
formulationvariations(Rahal1995;Elhayetal.2014)),andloop-basedmethods(Loopflow
correction(Cross1936)). Theirrelativeperformanceintermsofspeed,rate-of-convergence,
and accuracy depends among other things on the topology of the target network: size
of the forest component, the number of network loops, and the density of these network
loops. It is difficult to evaluate the impact of these topology factors by only examining
the incidence matrix that describes the pipe network connectivity. As a result, the best
method to use for a particular network cannot be easily determined a priori. Moreover,
extracomplexity isintroducedwhenamulti-runhydraulic assessmentisrequired. During
a multi-run hydraulic simulation, the elapsed computation time of each method can be
brokendownintotwoparts: thecomponentsthatareonlyrequiredtobeperformedonceat
theverybeginningforthesamenetwork,calledtheoverhead,andthecomponentsthatare
required to be carried out repeatedly for each separate run until the required number of
iterations has been met, called the hydraulic-phase. It is desirable to have a simulation
platform,giventhedifferentlevelsofrepetition,toimplementthesealternativealgorithms
efficiently. Equippedwithsuchaplatformauserwouldbeabletoeasilybenchmarkthe
performanceofalternativemethodsonasmallnumberofevaluationsforagivennetwork
and use that performance to inform the choice of algorithm to use for either a once-off
simulationsettingorforamultiplesimulationsetting(suchasforanevolutionaryalgorithm
(EA)).
ThisworkdescribesanextensibleWDSsimulationplatformcalledWDSLib. WDSLib
is a numerically robust, efficient and accurate C++ library that implements many WDS
simulation methods. WDSLib is written using a modular object-oriented design which
allowsuserstoeasilymixandinterchangesolutioncomponents,therebyenablingusersto
avoidredundantcomputations. Ithasbeenoptimizedtousesparsedatastructureswhich
areorientedtothepatternofaccessrequiredforeachsolutionmethod. WDSLibhasbeen
validatedforaccuracyonarangeofrealisticbenchmarkwaterdistributionnetworksagainst
referenceimplementationsandtestedforspeed. Theprogramacceptstheinputfileformats
ofthe industrystandardEPANET2 (Rossman 2000)toolkit andits performanceis faster
thanEPANET2inalltestedsettingsandbenchmarks.
Theremainderofthispaperisstructuredasfollows. Thenextsectiondescribesrelated
methodologies and implementations. A general description of the WDS demand-driven
steady-state problem is given in the next section. Section 3.6 presents a mathematical
formulation of the network and the solution methods that are used in WDSLib. The
tool-kitstructureisthengiveninsection3.7. Thisisfollowed,insection3.8,bythetoolkit
implementation details. Section 3.9 provides some examples of how the toolkit can be
utilized in a simulation work flow. The results are discussed in Section 3.10. Finally,
section 3.11 summarizes the results of this paper and describes future extensions to the
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toolkit.
3.5 Background
This section describes related water distribution system network solution methods and
implementations. The first sub-section describes solution methods, including those used
byWDSLib. This isfollowed by adescription ofcurrentlyavailableimplementations and
comparesthesewithWDSLib.
3.5.1 Related Methods
Thisresearchconsidersawaterdistributionmodelmadeupofenergyconservationequations
andthedemanddrivenmodelcontinuityequations. TheHardyCrossmethod(Cross1936),
also known as the loop flow corrections method, is one of the oldest methods and uses
successive approximations, solving for each loop flow correction independently. It is a
methodthatwaswidelyusedforitssimplicityatthetimewhenitwasintroduced. More
thanthreedecadeslater,EppandFowler(1970)developedacomputerversionofCross’s
methodandreplacedthenumericalsolverwiththeNewtonmethod,whichsolvesforall
loop flow corrections simultaneously. However, this method has not been widely used
because of the need (i) to identify the network loops, (ii) to find initial flows that satisfy
continuityand(iii)tousepseudo-loops.
The GGA is a range space method that solves for both flows and heads. It was the
first algorithm, in the field of hydraulics, to exploit the block structure of the Jacobian
matrixtoreducethesizeofthekeymatrixinthelinearizationoftheNewtonmethod. The
GGAhasgainedpopularitythroughitsrapidconvergencerateforawiderangeofstarting
values. This is the result of using the Newton method on an optimizations problem that
hasaquadraticsurface. However,itwasreportedbyElhayandSimpson(2011)thatthe
GGAfailscatastrophicallyinthepresenceofzeroflowsinaWDSwhentheheadlossis
modeled bythe Hazen-Williams formula. Regularization methodshave beenproposed by
bothElhayandSimpson(2011)andGorevetal.(2012)todealwithzeroflowswhenthe
headlossismodeledbytheHazen-Williamsformula.
TheGGAasitwasfirstproposed,appliedonlyfortheWDSsinwhichtheheadlossis
modeledbytheHazen-Williamsformula,wheretheresistancefactorwasindependentof
flow. Rossman(1994)extendedtheGGAtoallowtheuseoftheDarcy-Weisbachformula.
Ithasbeen pointedoutinSimpson andElhay(2010), however, thatRossmanincorrectly
treatedtheDarcy-Weisbachresistancefactorasindependentoftheflow. Theyintroduced
thecorrectJacobianmatrixtodealwiththis. Ithasbeendemonstratedthatoncethecorrect
Jacobian matrixis used, the quadratic convergence rate of theNewton method is restored.
Furthermore,ElhayandSimpson(2011)reportedthattheGGAdoesnotfailinthepresence
of zero flows when the derivatives of the Darcy-Weisbach Jacobian matrix are correctly
computedforlaminarflows.
The co-trees flow method (CTM) (Rahal 1995) is a null space method that solves
for the co-tree flows and spanning tree flows separately. The CTM, unlike the loop flow
correctionsmethod,doesnotrequiretheinitialflowstosatisfycontinuity. However,itdoes
require: (i)theidentificationoftheassociatedcirculatinggraph;(ii)thedeterminationof
the demands that are to be carried by tree branches; (iii) finding the associated chain of
branchesclosingacircuitforeachco-treechord;(iv)computingpseudo-linkheadlosses.
The reformulated co-trees flow method (RCTM) (Elhay et al. 2014) is also a null space
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because there are no clearly defined interfaces for the incorporation of third-party code
componentsinEPANET2,thereisnoguaranteethatindependentlyauthoredextensions
willbeeasytocombinewitheachother.
Intheabsenceofapopulareasy-to-modifyWDSsimulationplatformthereiscurrently
no straightforward means for comparing different solution methods. To date, when new
solutionmethodshavebeendevelopedtheyhavebeencomparedusingdifferentresearch
systems, on different platforms with different implementation languages. This leads to
difficulty in comparing methods, limits the reusability of code, and creates a barrier for
researchers to reproduce and replicate results. To address these issues, an extensible
framework is required that allows implementation of new methodologies to be easily
incorporatedwithoutanadverseimpactontheperformanceoftherestofthesystem.
To this end, a number of attempts have been made to implement an object-oriented
wrappertoencapsulatetheEPANET2solver(openNet(Morleyetal.2000)andOOTEN(van
Zyletal. 2003)). However,these twosystemswerefocusedonprovidingmoreflexibility
in the processing of input to the core EPANET solver. They did not address any issues
relatingtothe solutionprocess. CWSnet, aC++implementationinobject-orientedstyle,
was produced by Guidolin et al. (2010) as an alternative to EPANET 2.0. In CWSnet,
moreattentionhasbeen givento thehydraulicelements oftheWDSnetwork. Inaddition,
CWSNetprovidesapressuredrivenmodel,andtakesadvantageofthecomputingpowerof
the computer’s Graphics Processing Unit (GPU). However, in CSWnetthe data structures
representingthenetworkarespecializedtothesolutionmethodsthatituses. Thesedata
structuresarenoteasilyadaptedtoworkefficientlywiththedifferenttraversalorders,and
graph algorithms used by newlydeveloped solutionmethods. However, CWSnet still uses
thesamehydraulicsolverandthesamelinearsolvertechniquesimplementedinEPANET
2(Guidolinetal.2010).
Toaccommodatethedeficienciesreferredtoabove,thispaperpresentsanewhydraulic
simulationtoolkitWDSlib. WDSlibiscodedinC++,andincorporatesanumberofrecently
publishedtechniques. Thistoolkitoffersuserstheabilityto: (i)choosefrom,ormodify,
differentapproachesandimplementationsofdifferentWDSmodelanalyses,and(ii)extend
thetoolkittoincludenewdevelopments. Thesefeatureshavebeenimplementedusingfast
and modularized code. A focus ofattention in thisresearch has beenprogram correctness,
robustnessandcodeefficiency. Thecorrectnessofthetoolkithasbeenvalidatedagainsta
referenceMATLABimplementation. Thedifferencesbetweenallresults(intermediateand
final)producedbytheC++toolkitandtheMATLABimplementationwereshowntobe
smaller than 10 10. In the interest of toolkit robustness, special attention has been paid
−
to numerical processes to guard against avoidable failures, such as loss of significance
throughsubtractivecancellation,andnumericalerrors,suchasdivisionbyzero. Thedata
structures and code libraries in WDSLib are shared and all implementations have been
carefully designed to ensure fairness of performance comparisons between algorithms.
WDSLibusesapluggablearchitecturewheresolution-methods,andtheiraccompanying
pre-processing and post-processing code are easily substituted. In addition, different
numerical linear algebra techniques can be incorporated using a well-defined interface.
This concludes the discussion of related work. The mathematical formulations of the
solutionmethodsusedinWDSLibarepresentedinthenextsection.
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networkandthepipeindexesofthecorecomponentofthenetworkfromtheAlgorithm
1 (if the FCPA is used). In this algorithm, all water sources are the starting point of the
searchprocess,SN,andmarkedasvisited. ThenodesinSN arethenusedastoidentifya
spanning tree within the WDS. This is achieved by repeatedly finding all adjacent pairs,
nodetandpipes,ofandremovingthefirstnodeinSN byusingtheadjacencylist. Ifthe
adjacent node t is not visited then node t is inserted into the spanning-tree node vector,
STN, and search node vector, SN, and node t is marked as visited and pipe s to the
spanning-treepipevector,STP,andpipesismarkedasvisited. Iftheadjacentnodetis
visitedandthepipesisnotvisitedthenthepipesisinsertedintotheco-treepipevector,
CTP andmarkpipesasvisited. ThisprocessisrepeateduntilSN isempty. Theoverall
time-complexity of this algorithm is O(n + n ) (compared to O(n n ) as mentioned
p j p j
above)isthesameasthebestasymptoticcomplexityofbreadth-firstsearchonagraph.
3.9 Example Applications
WDSLib consists of a collection of functions which can be used either as a standalone
applicationforfastone-offsimulationsorasalibraryofsoftwarecomponentsthatcanbe
integratedintoauser’sownWDSsolutionprocesses. Thissectionpresentstwoexample
applications. The first application is the setup for a basic one-off simulation of a WDS.
Thesecondapplication(describedinsubsection3.9.1)presentsanexampleusingWDSLib
toimplementasimple1+1EvolutionaryStrategy(BeyerandSchwefel2002)(1+1-ESor,
morecommonly,1+1EA)forsizingpipesinaWDS.
Example 1 - Once-off Simulation
The setup for WDSLib as a standalone application is straightforward. The user provides a
configurationtextfilethatspecifiesinputandoutputfilenames;thenameofthesolver;the
desiredoutputvariables;andsimulationparameters. Thesevalueshavesensibledefaults
so the user can set up the solver by using a minimal configuration such as that shown in
Fig.3.4. Byusingthis configfile,WDSLibis configuredtorun asinglehydraulic analysis
ofthenetworkthatisstoredassay"hanoi.inp",anEPANET-formattedinputfile,under
"Network/"sub-directory,usingthereformulatedco-treeflowsmethodwiththeforest-core
partitioningalgorithm. Thefullsetofconfigurationparametersforonceoffsimulationsis
showninFig.3.10inAppendix3.16.
3.9.1 Example 2 - A Simple Network Design Application
AsaminimalistexampleoftheapplicationofWDSLibtoaWDSnetworkdesignproblem,
the following example uses 1+1EA for optimally sizing pipe diameters. This algorithm
takes an existing network with randomly generated pipe diameters and optimizes the
networkto minimizecost,subjectto givenpressurehead constraints. A 1+1EAisavery
simple evolutionary strategy (Beyer and Schwefel 2002) which starts with a randomly
generated individual (in this case a WDS diameter configuration). This 1+1EA then
progressesbyapplyingamutationtoarandompipediametersize,andthenevaluatingthe
new individual. If the new individual is better it replaces the old network. This process
continuesinaloopuntilagivennumberofevaluationsisreached.
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Thisconcludesthepresentationofexamplesinthiswork. Thenextsectionpresentsa
casestudythatillustratestheperformanceofWDSLibinamulti-simulationsetting.
3.10 Case Study
The following presents timing results for WDSLib running the 1+1EA described in the
previous section. The results below compare the four different solvers plus EPANET2.
Note, that detailed timings for once-off simulations comparing the four methods can be
foundinQiuetal.(2018). Threenetworkswerebenchmarkedintheseexperiments. These
were the N , N , and N case-study networks used in Simpson et al. (2012). Table 3.7
1 3 4
summarizesthecharacteristicsofthesenetworks.
Table3.7. Benchmarknetworkssummary
FullNetwork Forest&CoreNetworks Co-treeNetwork
Network n n n n (n /n#) n n n
p j s f f p jc pc ct
N 934 848 8 361(38%) 573 487 84
1
N 1975 1770 4 823(42%) 1152 947 205
3
N 2465 1890 3 429(17%) 2036 1461 757
4
Table3.8showstheresultsofthe1+1EAfromFig.3.5fortheGGA,GGAwithFCPA,
RCTM, RCTM with FCPA and the EPANET2 solvers. For each of the four WDSLib
solversabove,thetimingsare givenforrunningtheEAwithandwithouttheL1modules
hoistedoutthemainEAloop. EachexperimentevaluatestheWDSnetwork100,000times.
Andthebestperformingmethodforeachnetworkishighlightedinbold. Itisimportantto
notethat1+1EAusingboththeGGAandtheWDSLib
Table 3.8. The actual 1+1 Evolutionary Algorithm run-time with 100,000 evaluations
(min.) foreachofthefoursolutionmethodsappliedtonetworksN ,N ,andN
1 3 4
GGA GGAwithFCPA RCTM RCTMwithFCPA EPANET
min. min. min. min. min.
N 6.73 4.64 4.53 4.13 9.81
1
N 15.21 9.79 13.75 10.30 26.43
3
N 21.14 16.29 23.92 21.93 67.11
4
The results show that the EA runs using WDSLib are substantially faster than the
runs using the EPANET2 solver. This is, in part, due to the fact that the EPANET2
solveris designedasa standalonesolverwhichdoesnot facilitatelifting outofinvariant
computationsfromtheEAloop.
Asa demonstrationofhow theperformanceof anEA canbetraced Fig.3.9 showsthe
evolutionofthefitnessvaluesoftheN network. Thesetraceswereextractedfromafile
1
written to in line 9 in Fig. 3.8. As can be seen, the cost and the pressure head violation
terms drop during the EA run. Note that there will be considerable variation between
1+1EArunsduetoitshighlystochasticnature.
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4.2 Abstract
In recent years a number of new WDS solution methods have been developed. These
methods have been aimed at improving the speed and reliability of WDS simulations.
However,todate,thesemethodshavenotbeenbenchmarkedagainsteachotherinareliable
way. Thisresearchaddressesthisproblembyusinganewlydevelopedsoftwareplatform,
WDSLib, as a fair basis for a detailed comparison of the performance of these methods
under different settings. In this work, efficientimplementations of three solution methods,
the Global Gradient Algorithm (GGA), the forest-core partitioning algorithm (FCPA), and
thereformulatedco-treeflowmethod(RCTM),andcombinationsofthese,arecompared
on eight case study benchmark networks containing between 934 and 19647 pipes and
between 848and 17971nodes. These simulationswere carriedout underboth aonce-off
simulationsettingandamultiplesimulationsetting(suchasoccursinageneticalgorithm).
Timingsfor thesebenchmarkruns aredecomposed intostagesso thattheperformance of
thesemethodscanbeeasilyestimatedfordifferentsettings. Theresultsofthisstudywill
helpinformthechoiceofsolutionmethodsforgivencombinationsofnetworkfeaturesand
givendesignsettings. Inaddition,timingresultsarecomparedwithEPANET2.
4.2.1 Keywords
water distributionsystems solution;Forest-Core Partitioning Algorithm;Global Gradient
Algorithm;ReformulatedCo-treeFlowMethod;hydraulicanalysis;EPANET.
4.3 Introduction
Water Distribution Systems (WDSs) are frequently modeled by a system of nonlinear
equations,thesteady-statesolutionsofwhich,theflowsandheadsinthesystem,areused
in WDS design, management and operation. In a design setting, the solutions might be
used as part of an optimization problem to determine the best choices of some network
parameterssuchaspipediameters. Inamanagementsetting,thesolutionsmightbeused
for the calibration of network parameters such as demand patterns. In an operational
environment, new solutions might be needed to adjust control device settings whenever
newsupervisorycontrolanddataacquisition(SCADA)informationbecomesavailable.
ThemostwidelyusedWDSsimulationmethodincurrentuseistheGlobalGradient
Algorithm(GGA)(TodiniandPilati1988),whichsolvesthenon-linearsystemofequations
representingtheWDS.TheGGAanditsimplementationsexhibitexcellentconvergence
characteristics for a wide range of starting values and a wide variety of WDS problems.
However, some networks have structural properties which can be exploited to further
improvetheefficiencyofthesolutionprocess. TheGGA,arangespacemethod,exploits
theblockstructureofthefullJacobianmatrixinordertoproduceasmallerkeymatrixinthe
linearizationoftheNewtonmethod. Thereformulatedco-treeflowsmethod(RCTM)(Elhay
etal.2014),anull-spacemethod(Benzietal.2005),canfurtherexploittheblockstructure
of the Jacobian matrix to produce, in realistic WDSs, an even smaller key matrix. This
is achieved by dealing separately with the spanning tree and the co-tree in the Newton
methodlinearization.
AnotheravenueforreducingcomputationcanbeexploitedbyusingtheForest-Core
Partitioning Algorithm (FCPA) (Simpson et al. 2012) to separate the problem into its
linearandnon-linearcomponents. TheobservationunderpinningtheFCPAisthatmost
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Methods
WDSs have trees, the collections of which are called forests. The complement of the
forestinanetworkiscalledthecore. Theflowsinaforestcanbecomputeda-prioribya
linear process. Hence, the dimension of the key matrices in the solution process can be
significantlyreducedwhentheforestisalargepartofthenetwork.
Withthedevelopmentofdifferentsolutionmethods,WDSsimulationpackageusersare
facedwithachoiceofwhichsolutionmethodormethodstoapply. Previouspublications
performedcasestudiescomparingtheperformancesoftheirrespectivemethodstotheGGA.
However, thesecomparisonswereoften doneusingdifferent implementationlanguages,
and different levels of code optimization – which makes cross-comparison of methods
difficult. Consequently, there is a need for a study which reliably compares the relative
performanceofthesemethods usingafast,carefully designedcodeimplementation. To
thisend,thisworkpresentsathoroughbenchmarkstudytocomparetheperformanceof
GGA,GGA-with-FCPA,RCTM,andRCTM-with-FCPAforarangeofcasestudynetworks
usingafastC++implementation. Thetimingsfortheserunsaredecomposedaccording
tohowofteneachsolutioncomponentisexecutedindifferentsimulationsettings. From
thesetimingsitispossibletoaccuratelypredictruntimesforlong-runmultiplesimulation
settings. Toconfirmtherelevanceoftheseresults,thetimingshavebeencomparedwith
thespeed ofthe industrialand research standard toolkitof EPANET2 (Rossman2000) and
wasfoundtobefasterinallcases.
Thispaperisorganizedasfollowed. Adetailedreviewofexistingsolutionmethodsis
given in the next section. The section following presents the mathematical formulation
of each method. The motivation for a benchmark study is then given, followed by the
methodology used in this paper to carry out a benchmark study. The description of the
module categorization is then presented. This is followed by a case study of the four
solutionmethods appliedtotheeight casestudynetworks. The resultsarediscussedin the
nextsection. Thelastsectionofferssomeconclusions.
4.4 Literature Review
This section provides a review of the algorithms that are tested in this paper. A brief
developmenthistoryofWDSsolutionalgorithmsispresentedinthefirstsubsection. The
nextsubsectiongivesanoverviewoftheGGAanditsdevelopment,followedbyanoverview
of solution methods which use the null space approach (such as co-trees flow method
(CTM) and RCTM). Finally, a review of the methods that use graph theory to simplify
problemcomplexityarepresented.
4.4.1 Development history of the WDS algorithms
Thisresearchconsidersawaterdistributionmodelmadeupofenergyconservationequations
andthedemanddrivenmodelcontinuityequations. TheHardyCrossmethod(Cross1936),
also known as the loop flow corrections method, is one of the oldest methods and uses
successive approximations, solving for each loop flow correction independently. It is a
methodthatwaswidelyusedforitssimplicityatthetimewhenitwasintroduced. More
thanthreedecadeslater,EppandFowler(1970)developedacomputerversionofCross’s
methodandreplacedthenumericalsolverwiththeNewtonmethod,whichsolvesforall
loop flow corrections simultaneously. However, this method has not been widely used
because of the need (i) to identify the network loops, (ii) to find initial flows that satisfy
continuityequationand(iii)tousepseudo-loops.
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4.6.3 Network Partitioning
Associated with a WDS is a graph G=(V, E), where the elements of V are the nodes
(vertices) of the graph G and elements of E are the pipes (links) of the graph G. In this
subsection, the permutation of the system equations (4.3) for the FCPA is introduced,
followedbyadescriptionoftheRCTM,whichfurtherexploitstheblockstructureofthe
Jacobianmatrix.
Forest-CorePartitioningAlgorithm
Inademand-drivenmodel,itispossibletoexploitthefactthateveryWDScanbedivided
intotwosubgraphs: atreedsubgraph(forest)G = V ,E andaloopedsubgraph(core)
f f f
G = (V ,E ),sothatE E = E,E E = (cid:16),V V(cid:17) = V. Allflowsandheadsin
c c c f C f C f C
∪ ∩ ∅ ∪
boththeforestandthecoremustbefound. Theflowsintheforestcanbefoundbyalinear
process before the iterative solution phase and theheads in the forest can be found linearly
aftertheiterativephase.
Simpson et al. (2012) proposed the FCPA, which partitions the network into a treed
component and a looped component (referred to as the core) thereby reducing the com-
putationtimewherethenetworkhasasignificantforestcomponent. TheFCPAstartsby
generatingapermutationmatrix
n n
p j
n S O
f
n P O
P = pc (4.9)
1 n O C
jc
n O T
f
S
, where Rnp np is the square orthogonal permutation matrix for the pipes,
"P # ∈ ×
S Rn f np is the permutation matrix which identifies the pipes in the forest as distinct
×
∈
fromthoseofthecoreoftheWDS,P Rnpc×np isthepermutationmatrixforthepipes
∈
C
in the core of the WDS, Rnj nj is the square orthogonal permutation matrix for
"T # ∈ ×
the nodes, C Rnjc×nj is the permutation matrix for the nodes in the core of the WDS,
∈
T Rn f nj isthepermutationmatrixwhichidentifiesthenodesintheforestasdistinct
×
∈
fromthoseofthecoreoftheWDS.
Anewlemmaisproposedasfollows:
LEMMA1. Suppose
n
m P
Q = 1 ,
m S
2 !
Q Rn n, is an orthogonal permutation matrix and that D = diag d ,d , ,d
× 1 2 n
∈ { ··· } ∈
Rn n isdiagonal. Then
×
PDST = 0 (4.10)
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where: R = K A RT ; R = K A RT ; L = R R T ; F(m) = K F(m)K T ;
1 1 1 2 2 1 21 − 2 −1 1 1 1
F(m) = K F(m)K T G(m) = K G(m)K T ; G(m) = K G(m)K T ; a = K A e ;
2 2 b2 1 1 b 1 2 2 2 1 b1 2 l
a = L K A e +K A e ; W(m) = L (F(m) ) 1L T + (F(m) ) 1. Note that in
2 21 1 b2 l 2 2 l b 21 1 − 21 b 2 − b
Eq. (4.20), an initial set of the co-tree flows
q(0)
is needed to commence the solution
2
b b
process.
The heads are found after the iterative process of the RCTM by using a linear solution
process:
R h = F q(m+1) (F G )q(m) a (4.22)
1 1 1 − 1 − 1 1 − 1
Thispartitioningofthenetworkequationsreducesthesizeofthenon-linearcomponent
ofthesolverton n (thenumberofco-treeelementsinthenetwork). Ithasbeenproven
p j
−
by Elhay et al. (2014) that the RCTM and the GGA have identical iterative results and
solutionsifthe same startingvaluesareused. However,forRCTM,theuseronlyneeds to
set the initial flow estimates for the co-tree pipes,
q(0)
, in contrast to GGA where initial
2
flow estimates are required for all pipes. The flows in the complementary spanning tree
pipesaregeneratedbyEq.(4.20).
4.7 Methodology
This section describes the methodology used to carry out a comparative study of the
WDS solution methods. The following describes the software platform used to run
the benchmarking simulations. This description is followed by the proposed algorithm
evaluationmethod.
4.7.1 The Software Platform
Torunthebenchmarktestsrequiredbythisstudyahydraulicsimulationtoolkit,WDSLib,
wascreated. Thistoolkit,writteninC++,incorporatedthesolutionmethodsstudiedinthis
paper,whichincludetheGGA,theGGAwiththeFCPA,theRCTM,andtheRCMTwith
theFCPA.Inordertoprovideausefulplatformforcomparison,thesolutionmethodswere
implemented using fast and modularized code. A focus of attention in this research has
beentheimplementationcorrectness,robustnessandefficiency. Thecorrectness∗ ofthe
toolkithasbeenvalidatedagainstareferenceMATLABimplementation. Thedifferences
betweenallresults(intermediateandfinal)producedbytheC++toolkitandtheMATLAB
implementation were shown to besmaller than 10 10. Inthe interestof toolkitrobustness,
−
specialattentionhasbeenpaid tonumericalprocessestoguardagainstavoidablefailures,
suchaslossofsignificancethroughsubtractivecancellation,andnumericalerrors,such
as division by zero. The data structures and code libraries in the toolkit are shared and
all solution method implementations have been carefully designed to ensure fairness of
performancecomparisonsbetweenalgorithms.
The following subsections describe the measures taken in the implementation the
solutionmethodstohelpensurethevalidityofthetimingexperimentsforthecasestudy
results. Theseincludemeasurestoensureaccuratetimingresults,minimizationofmemory
use,andnumericalrobustness.
∗termsrecognizedinComputerSciencewillbedesignatedbyasterisksuperscript
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RCTM,and(iii)reducethenumberofpipesinthespanningtree. Thiscanbeseenbythe
per-iterationexecutiontimesforeachoftheL modules,whichareshownintheTable4.6.
3
Table 4.7. The number of iterations required for each of the four solution methods to
satisfythestoppingtestfortheeightcasestudiesnetworks. The"relativediff."referstothe
relativedifferencecomparedtothenumberofiterationsfortheGGA
GGA GGAwith RCTM RCTMwith Relativediff.
FCPA FCPA usingRCTM
N 8 8 12 12 +50%
1
N 8 8 13 13 +62.5%
2
N 8 8 9 9 +12.5%
3
N 9 9 13 13 +44.4%
4
N 8 8 10 10 +25%
5
N 10 10 12 12 +20%
6
N 9 9 13 13 +44.4%
7
N 9 9 11 11 +22.2%
8
Thenumberofiterationsrequiredforeachofthefoursolutionmethodstosatisfythe
stopping test for the eight case studies networks is shown in the Table 4.7. It is evident
from Table 4.7 that the GGA took exactly the same number of iterations to satisfy the
stopping test with or without the FCPA. The flows in the forest network satisfy a linear
system,whichdoesnotchangefromoneiterationtothenext. Therefore,theflowsinthe
forestpipesreachtheirsteady-stateafterthefirstiteration. Similarly,theRCTMwithor
without FCPA takes the same number of iterations. In the cases that were analyzed in
this study, the RCTM required a greater number of iterations to satisfy the stopping test
comparedto theGGA.Thisis becausedifferentmechanisms areusedto generate asetof
initialflowsforthetwomethodsasdiscussedpreviously.
ItisworthusingtheFCPAinconjunctionwithboththeGGAandRCTMforaonce-off
simulation given that FCPA decreases the L per-iteration time without increasing the
3
numberofiterationspermodule. Interestingly,asmallerper-iterationtimeisrequiredby
theL modulesoftheRCTMexceptfornetworkN . However,RCTMrequiresagreater
3 8
numberofiterationsforallthecasestudynetworks. Thissometimescausesagreatertime
fortheRCTMtosatisfythestoppingtest.
4.9.2 Multiple Simulation Setting
The performance of the four solution methods under the multiple simulation setting are
compared. Pipediametersfortheeightcasestudynetworkswererandomlygeneratedat
eachevaluationtosimulateanevolutionaryalgorithmrun. Itisimportanttonotethatthe
useof randomly generated pipediametersgivesan overestimateof thetotalruntime. This
is because, as EA’s progress, the pipe diameters in its population become increasingly
realistic,which,onaverage,shouldreducethenumberofiterationsattheL level.
3
Table4.8 and Table 4.9 show thedetailed timing resultsof multiple simulationswith
number of evaluations N = 100,000 for each of the four solution methods applied to
E
thenetworksN andN . Table4.8showsthatexploitingthetreednatureofnetworkN
1 8 1
givestheFCPAa29%timesavingovertheGGAand15%timesavingovertheRCTM.A
smallersavingisachievedbytheuseoftheFCPAfornetworkN : 14%fortheGGAand
8
9%fortheRCTM.Inamultiplesimulationsetting,theRCTMismoretiming-consuming
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Publication 3: A Bridge-Block Partitioning
Algorithm for Speeding up Analysis of
Water Distribution Systems
5.1 Synopsis
In Chapter 4, WDSLib, a waterdistribution system simulation toolkit thatwas developed
inChapter3,wasusedasafairbasisforadetailedcomparisonoftheperformanceoffour
waterdistributionsystemsolutionmethods,namelytheglobalgradientalgorithm(GGA),
the GGA with the forest-core partitioning algorithm (FCPA), the reformulated co-tree
flowsmethod(RCTM),andtheRCTMwiththeFCPAunderdifferentsettings. Another
typeofgraphproperty,bridgeandblockcomponents,hasbeeninvestigatedinthischapter.
The bridge-block partitioning algorithm (BBPA)begins by using theFCPA toseparate the
forest component from the core component. Then, the BBPA further partitions the core
componentofthenetworkintoblockandbridgecomponents.
Bridgecomponentsarethepipesinthecorethatarenotpartofanyloop. Thesolutions
for thebridge componentscanbe foundbya linearprocess– inthesame way ascanthe
forestcomponent intheFCPA.Theremainder ofthenetworkisconsistingofblocks and
solutionsfortheseblockcomponentscanbefoundseparately. Itispossibletoseparatetwo
blockswithasinglenodecalledacut-vertex. Theadvantagesinspeedandreliabilityfor
the BBPA arise, in part, fromthe smaller systems thatresult from partitioning thenetwork
intothesesmallerblocks,ifthecorecomponentoftheWDSgraphisone-connected.
TheBBPAexploitsthefactthattheflowsandheadsinoneblockcomponentareweakly
coupledwiththoseoftheotherblockcomponentsandthesolutionoftheflowsandheads
in a bridge component is a linear process. The convergence rate for the solution of the
core component of a WDS, without the BBPA, is restricted to that of the worst block of
the network. The number of iterations required by each block is bounded above by that
requiredbytheunpartitionedsystem.
The use of BBPA can also improve the reliability of the solution. The numerical
reliability of the solution can be determined by the condition number of the Schur
complement. Theconditionnumberofamatrixistheratioofthelargesttothesmallest
singular value of any square matrix. In most cases, the condition numbers for all the
individualblockswillbesmallerthantheconditionnumberofthefullmatrix.
In this Chapter, the advantage of using BBPA is demonstrated on eight case studies
withbetween932to19647pipesandbetween848and17971nodes. Theglobalgradient
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5.2 Abstract
Manywaterdistributionsystem(WDS)solutionmethodshavebeendevelopedtoperform
demand-driven steady-state analysis. These methods are used to solve the non-linear
systemofequationsthatmodelaWDS.WDSnetworkshavestructuralpropertiesthatcan
oftenbeexploitedtospeedupthesesolutionmethods. Onesolutionmethodthatexploits
these structural propertiesis theforest-core partitioning algorithmthat was proposedas
a pre-processing and post-processing method that can be used to separate the network
into a linear forest component and a non-linear core component. This paper presents a
complementarymethodforpre-andpost-processingcalledthebridge-blockpartitioning
algorithm (BBPA). This method further partitions the core component of the network into
anumberoflinearbridgecomponentsandanumberofnon-linearblockcomponents. The
use of BBPA to partition a WDS network provides significant advantages over current
solutionmethodsintermsofbothspeedandsolutionreliability.
5.2.1 Keywords
Global gradient algorithm (GGA); Graph Theory; Bridge-Block Partitoning; Water
distributionsystems;Hydraulicanalysis.
5.3 Introduction
Hydraulic simulation algorithms use mathematical models designed to simulate the
hydraulic performance of a water distribution system (WDS) and have played a critical
roleinthedesign,operation,andmanagementofWDSsinresearchandindustry. These
models have been used for (1) optimizing WDS network design parameters (such as
pipe diameters), (2) for calibrating network parameters (such as demand patterns), (3)
conductingreal-timemonitoringandcalibrationofthenetworkelementsinasupervisory
controlanddataacquisition(SCADA)operationalsetting,and(4)adjustingcontroldevices
(suchasvalves). Inhydraulicsimulation,thesystemofequationscanbeformulatedasa
large andsparsenon-linear saddle-pointproblem. Thereare severalwell-knowniterative
methods for solving the non-linear saddle-point problem. These include: range space
methods,nullspacemethods,andloop-basedmethods.
ThemostwidelyusedWDSsolutionmethodistheGlobalGradientAlgorithm(Todini
andPilati1988). TheGGA,arangespacemethod,takesadvantageoftheblockstructureof
thefullJacobianmatrixtoachieveasmallerkeymatrixinthelinearizationoftheNewton
method. Since the development of the GGA, numerous new WDS hydraulic solution
methodshavebeenproposedandimprovementshavebeenmadetoexistingWDShydraulic
solution methods. Most of these new WDS hydraulic solution methods employ graph
theorytodecomposeorpartitiontheWDSnetworkgraphintosub-graphswhichresultsin
a smaller system of equations. Deuerlein (2008) introduced a decomposition model for
a WDS network graph, in which the one-connected components are categorized as the
forest componentandthebiconnectedcomponents arecategorizedasthe corecomponent.
After removing the forest component, the core component can be further partitioned into
blocks that are connected by bridge elements. After the partitioning processes, a loop
flow corrections method is then used. Simpson et al. (2012) proposed a matrix based
identification method for the forest component and the core component and introduced
theforest-corepartitioningalgorithm(FCPA).IntheFCPA,flowsandheadsintheforest
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(a)ExampleA (b)ExampleB
Fig.5.1. Twoexamplenetworksofblocks,bridges,andcut-vertices
illustrated in Fig. 5.1(b). The node (cut-vertex 2) is a cut-vertex that separates the two
blocks. Thesetwoblockscanalsoalsobesolvedseparately,aswasthecaseinpart(a)of
theexample. Theadvantagesin speedand reliabilityfor theBBPA arise, inpart, from the
smallersystemsthatresultfrompartitioningthenetworkintothesesmallerblocksifthe
corecomponentoftheWDSgraphisone-connected.
The BBPA exploits the fact the flows and heads in one block component are weakly
coupledwiththeseoftheotherblockcomponentsandthesolutionoftheflowsandheads
inabridgecomponentisalinearprocess. Theconvergencerateforthesolutionofthecore
component of a WDS, without the BBPA, is restricted to that of the worst block of the
network. Solvingeachblockseparatelyreduces thenumberofiterationsexecutedtothe
numberofiterationsrequiredbythatblock.
There is a number of advantages to using the BBPA to identify the linear bridge
componentsandtheblockcomponentsofaWDSnetwork:
1. The number of iterations required by each block is bounded by that required by
the unpartitioned system – solving the flows and heads in each block separately
significantly reduces the overall computational time for the non-linear solver in
almostallcases.
2. It improves the numerical reliability of the solution. The numerical reliability of
thesolutioncanbedeterminedbytheconditionnumberoftheSchurcomplement.
The condition number of amatrix is the ratio of the largest to the smallest singular
valueofanysquarematrix. Aroughruleofthumbis: onedigitofreliabilityinthe
solution is lost for every power of ten in the condition number. If a square matrix
is partitioned into block diagonal form by orthogonal permutations, the condition
numbers ofblockscan be nogreater than that ofthe full matrix. Inmost cases, the
condition numbers for all the individual blocks will be smaller than the condition
numberofthefullmatrix. Thisphenomenonisillustratedlaterinthispaper.
3. Itreducestheneedtoregularizeforthepresenceofzeroflows(ElhayandSimpson
2011). IthasbeenpointedoutbySimpsonetal.(2012)thatsolvingfortheflowsand
heads separately can avoid thenumerical failure that occurswhen there are nodes
withzerodemandpresentintheforest. Itisshowninthispaperthatthereareblocks,
in some networks, that have zero accumulative demands. The solutions of these
networksneedaregularizationmethodto dealwiththepresence ofthezeroflows
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5.5.2 The properties of the system of equations after bridge-block
partitioning
In the BBPA, a full WDS network is partitioned into n smaller independent non-
b
linear systems by permuting the original full system of equations using two orthogonal
permutationsP andR. Oneofthemaincontributionsofthispaperistoshowthattheuse
of the BBPA can significantly reduce the computational loads and improve the numerical
reliabilityoftheresults.
TheBBPAcanbeusedtoimprovethereliabilityofsolutionoftheloopedcomponentin
thefinalWDSsolution. Thisisbecausetheconditionnumber,theratiobetweenthelargest
tothe smallest singularvalue ofamatrix,can beusedto estimate thelossofreliable digits
in solving a linear system with that matrix. The orthogonal permutations of the BBPA
shufflethen singularvaluesoftheSchurComplementintotheircorrespondingblocks.
j
This is because pre-and-post-multiplying a matrix by orthogonal matrices preserves the
singularvalues. Theupperboundof thelargestsingularvalueofall blocksisthe largest
singularvalueofthefullsystemandthelowerboundforthesmallestsingularvalueofall
blocksisthesmallestsingularvalueforthefullsystem. Therefore,theconditionnumber
ofeachblockat thesolutionisboundedabovebytheconditionnumber ofthefullsystem
of equationsbut in mostcases will besmaller. Moreover, theonly occasions whenone of
theblockshasthesameconditionnumberasthefullsystemiswhereboththehighestand
lowestsingularvaluesarepresent inthesameblock. Eveninthis particularcasetheother
blocksinthesystemwillhavelowerconditionnumbersthanthefullsystem.
Furthermore,theuseoftheBBPAcanminimizetheneedtouseregularizationmethods
forhandlingzero-flows. IntheFCPApaper(Simpsonetal.2012),theauthorspointedout
thatitiscommonforzeroflowstooccurattheendsoftreeswithzerodemands. Similarly,
it is also possible for all nodes in the end blocks to have zero demands. The GGA fails
catastrophicallyattheseblockswhentheheadlossismodelledbytheHazen-Williamhead
loss model. One side-effect of identifying these end blocks with zero nodal demands is
zeroflowscanbeassignedtoallpipesintheseblocksandtheheadofpseudo-sourcecanbe
assignedtoallnodesintheseblocks. Whenzeroflowsoccurinotherblocks,regularization
isneededonlyfortheblockswiththepresenceofzeroflowsinsteadofthefullsystem.
Inadditiontotheimprovementofthenumericalreliabilityofthefinalresult,theuseof
theBBPAcansignificantlyreducecomputationalloads. Thisreductionincomputational
loads is achieved through: (1) the bridge component being solved by a linear process,
the removal of which reduces the number of non-zeroes in Schur component, (2) the
probablereductioninthenumberiterationsrequiredbyeachblockasshowninAppendix
insection5.13,and(3)thenon-linearsystemofequationsforeachblockisindependentof
otherblockswhichallowseachblocktobesolvedinparallel.
5.6 Bridge-Block Partitioning Algorithm
ThestepsoftheBBPAarenowdescribed. TheBBPAstartswithaforestsearch algorithm
toidentifytheforestcomponentasdistinctfromthecore. Thisisfollowedbyidentifying
all the blocks and bridges in the core, and updating the demands for the cut-vertices by
using Stage 3 as givenbelow, a variation of the algorithm detailed by Hopcroft and Tarjan
(1973). Notethatthisalgorithmisbasedonthedepth-firstsearchandrunsinlineartime.
Therearetwowaystosolvethecoreofthenetwork: inparallelorserially.
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Thesystemofpipeheadlossandnodalcontinuityequationsfortheexamplenetworkis
G1 1 0 0 0 0 0 q1 el7
G2 −1 0 0 0 0 1 q2 0
G3 G4 − 01 −1 1 0 1 0 0 0 0 0 0 q q3 4
0 0
G5 G6 0 0 −0 1 1 0 − 11 0 0 0 0 q q5 6
0 0
G7 G8 0 0 − 01 00 00 −1 1 10 q q7 8 = 0 0 . (5.22)
1 0
0
− 0 01 − 1 01 −0 11 0 10 −0 01 −0 01 000
h h h1 2 3
d d d1 2 3
0 0 0 0 0 0 0 0 − 01 01 10 −0 1 h h4 5
d d4 5
0 1 0 0 0 0 0 1 h6
d6
By permuting the rows (pipes) in the ordering given by p = 1;2;3;7;8;4;5;6 and
{ }
thecolumns(nodes)intheorderinggivenbyv = 1;6;2;5;3;4 ,thesystemofequations
{ }
inEq.(5.22)canberearrangedintothefollowingblockstructure:
Pipes Nodes
Block B B B B B B
1 2 3 1 2 3
B 1 G 1 1 0 0 0 0 0 q 1 e l7
G 2
G
−1
1
1
0
0
1
0
0
0
0
0 0q q2 3 0 0
B B B2 3
1
1 1
13 G 07 G 08 G 04 G 05 G 06 − 0 0 0 0 0 0 1 0 0 0 − − −1 1 10 0 −11 0 0 0 0 0 1 1 0 −10 0 0 1
hq q q q q7 8 4 5 6 1
=
d0 0 0 0 0 1
(5.23)
B B2 3
00 0 0 0 − 01 0 0 0 − 10 0 0 0 −0 1 0 01 −01 0 01 −0 0 1 01 −00 0 1 1 −0 0 0 11 h h h h h6 2 5 3 4 d d d d d6 2 5 3 4
*theboldnumbersinthematrixrepresentthecut-vertices
Eq.(5.23) hasthree graphblocksas shown inFig. 5.2include Block1 (abridge), Block
2,andBlock3. Notethat,forcross-referencingpurposes,thisequationhasbeenlabeled
withtheblocknumbers(affiliatedwithpipesandnodes)correspondingtoeachentityin
theexamplenetwork. Thecut-vertices(cv andcv inFig.5.2)arehighlightedinbold
1 2
intheircorrespondingmatrixblocks. Intheequation, itisevidentthat thepermuted A
1
matrix is a block three by three, lower block triangular matrix which represents a WDS
withthethreegraphblocks(B ,B ,andB ).
1 2 3
The end block (B in Fig. 5.2) is a sub-network consisting of three pipes {4; 5; 6},
3
two nodes {3; 4}, and a pseudo-source at node {2}. The nodal demands of this block
do not need to be updated because this is the end block. The head of the node 2 (cv ) ,
2
whichisthecut-vertexbehavingasthepseudo-sourceforthisblock,canbemovedtothe
right-hand-sideof systemofequations usingEq.(5.16). Thesolution ofblock B canbe
3
foundseparatelyaftertheheadofthepseudo-sourceatnode{2}isfound.
The second block diagonal row (B in Fig. 5.2) is a sub-network consisting of four
2
pipes {2; 3; 7; 8}, three nodes {2; 5; 6}, and one pseudo-source at node {1}. This is
an intermediate block so that the demand at the node 2 (cv ), a cut-vertex that is not a
2
pseudo-source, needstobe updatedbyincreasingits demandbythesum ofdemands atall
nodesofitschildblock(B )asfollows: d = d +d +d usingEq.(5.20). Node1(cv ),
3 2 2 3 4 1
b
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whichisthecut-vertexbehavingasthepseudo-sourceforthisblock,B ,canbemovedto
2
the right-hand-side ofsystem ofequations using Eq. (5.16). The solution ofblock B can
2
befoundseparatelyaftertheheadofthepseudo-sourceatnode{1}isfound.
Finally,therootblock(B inFig.5.2)isasub-networkconsistingofpipe{1},node
1
{1},andsource{7}. BlockB isabridgecomponent. Thebridgecomponentcanbesolved
1
by using a linearprocess. The demandfor thenode 1 inFig. 5.2 (cv ), acut vertex inthe
1
rootblock,isupdatedbyincreasingitsdemandbythesumofdemandsatallnodesofits
childblock(B )asfollows: d = d +d +d +d +d +d andtheelevationheadfor
2 1 1 2 3 4 5 6
thesource stays thesame. Afterupdating thedemands andheads, the system ofequations
b
inEq.(5.23)becomes:
Pipes Nodes
Block B1 B2 B3 B1 B2 B3
B1 G1 1 0 0 0 0 0 q1 el7
BB B B B2 3 1 2 3 1 00 0 0 0 G 0 01 0 0 02 G 0 10 0 0 03 −G 0 0 1 0 017 −G 00 1 0 018 G 0 00 0 1 04 −G 0 00 0 1 15 G 0 0 10 0 06 0 0 0 0 0 0 0 1 0 0 1 0 0 0 −0 1 0 0 0 01 −10 0 0 0 01 0 0 0 0 1 1 0 −0 0 0 0 0 11 h h h h h hq q q q q q q2 3 7 8 4 5 6 1 6 2 5 3 4 = d1+d2 d+ 2d +3h h h h d d dd d+0 0 0 6 3 43 51 1 2 2 d +4 d+ 4d5+d6 (5.24)
Note that the system of equations obtained in Eq. (5.24) is equivalent to performing
blockGauss-JordaneliminationonEq.(5.23). Solvingthesystemofequationsinthisway
requires solving each block in a particular sequence, from the rootblock (B ) to the end
1
block (B ). The sequence that is required in the example network in Fig. 5.2 is: (1) to
3
findthesolutionofblockB ,therootblock;(2)tofindthesolutionofblockB usingthe
1 2
headofthenode one,cv ,inblockB ;and(3) tofindthesolutionofblockB ,theend
1 1 3
block,usingtheheadofthenodetwo,cv ,inblockB .
2 2
Furthermore,thesecondpipehead-lossblockequationorthesecondblockequation
(B )inEq.(5.24)is:
2
G q B h = B h ,
b 2 b 2 − 22 b 2 21 b 1
whichexpandsto:
G q 1 0 0 h
2 2 h 1
G q 0 1 0 6 h
3 3+ h = 1, (5.25)
G q 0 1 1 2 0
7 G q7 1 − 0 1 h 5 0
8 8 −
theright-hand-sideofwhichcanberewrittenas:
B h = B [v h ], (5.26)
21 b 1 − 22 3 1
whichexpandsto:
h 1 0 0
1 h
h 0 1 0 1
1 = h
0 0 1 1 1
− h 1
0 1 0 1
−
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usingEq.(5.21). SubstitutingitbackintoEq.(5.25),weget:
G q B h = B [v h ],
b 2 b 2 − 22 b 2 − 22 3 1
whichexpandsto:
G q 1 0 0 1 0 0
2 2 h h
G q 0 1 0 6 0 1 0 1
3 3+ h = h ,
G q 0 1 1 2 0 1 1 1
7 G q7 1 − 0 1 h 5 1 − 0 1 h 1
8 8 − −
whichcanfurthersimplifiedinto:
G q B [h +v h ] = O,
b 2 b 2 − 22 b 2 3 1
whichexpandsto:
G q 1 0 0
2 2 h h
G q 0 1 0 6 − 1
3 3+ h h = O.
G q 0 1 1 2 − 1
7 G q7 1 − 0 1 h 5 −h 1
8 8 −
Thethirdpipehead-lossblockequationorthethirdblockequation(B )inEq.(5.24)is:
3
G q B h = B h ,
b 3 b 3 − 33 b 3 32 b 2
whichexpandsto:
G q 1 0 h
4 4 h 2
G q + 1 1 3 = 0 . (5.27)
5
G
q5
0
−
1
h 4!
h
6 6 2
Eq.(5.27)canbefurthersimplifiedto
G q 1 0
4 4 h h
G q + 1 1 3 − 2 = O
5 G q5 0 − 1 h 4 −h 2!
6 6
usingasimilarmanipulationasforBlock2above.
Finally,thesystemofequationsinEq.(5.24)mayberewrittenas:
Pipes Nodes
Block B1 B2 B3 B1 B2 B3
B1 G1 1 0 0 0 0 0 q1 el7
BB B B B2 3 1 2 3 1 00 0 0 0 G 0 01 0 0 02 G 0 10 0 0 03 −G 0 0 1 0 017 −G 00 1 0 018 G 0 00 0 1 04 −G 0 00 0 1 15 G 0 0 10 0 06 0 0 0 0 0 0 0 1 0 0 1 0 0 0 −10 1 0 0 0 0 −110 0 0 0 0 0 0 0 0 1 1 0 −10 0 0 0 0 1 hh hh h52 46 3hq q q q q q q −− −− −2 3 7 8 4 5 6 1 hh hh h11 21 2 = d1+d2 d+ 2d +3 ddd d d+0 0 0 0 0 0 0 536 3 4d +4 d+ 4d5+d6 (5.28)
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5.7.2 Solving the example network
Considerthe network shownin Fig.5.2 andits permutedsystem ofequations, Eq.(5.28).
Eachblockbecomesanindependentsystemandcanbesolvedsequentiallyfromtheroot
blocktotheendblock. Thesystemofequationsfortherootblock,B (Block1inFig.5.2),
1
whichalsorepresentsabridge,is:
G 1 q e
1 1 = l 7 , (5.29)
1 0 h d +d +d +d +d +d
! 1! 1 2 3 4 5 6!
the solutionof whichcan beused tosolveits childblock, block B (Block 2in Fig.5.2)
2
byusing:
G 1 0 0 q 0
2 8
G 0 1 0 q 0
3 7
G 0 1 1 q 0
7 − 3
G 1 0 1 q = 0 , (5.30)
8 2
−
1 0 0 1 h 6 h 1 d 6
−
0 1 1 0 h h d +d +d
− 2 − 1 2 3 4
0 0 1 1 h 5 h 1 d 5
− −
andfinally,theendblock,blockB Block3inFig.5.2)canbesolvedbyusing:
3
G 1 0 q 0
4 6
G 1 1 q 0
5 5
−
G 0 1 q = 0 . (5.31)
6 3
1 1 0 h h d
3 2 3
−
0 1 1 h 4 h 2 d 4
− −
Thesystemsofequationsforeachofthethreeblockscanalsobesolvedinparallel.
Notethat,whenusingBBPA,iftheheadlossoftheexamplenetworkshowninFig.5.2
is modeled by the Hazen-William formula and the nodal demands at nodes three and
four are zero, this does not cause a failure of the method due to singularity of the Schur
complement,unliketheGGAandRCTMonthesamenetwork(ElhayandSimpson2011).
Inaddition,theblockwithzerototaldemandcanbesolved(1)priortotheiterativephase
by assigningzero flowstoall applicablepipesand (2)byassigningthe headsof thesource
toallnodesinthisblockaftertheiterativephase.
5.8 Relation of BBPA to other solution methods
TheBBPAcanbedescribedasapre-and-post-processingmethodforthefollowingreasons:
(1)itfindstheblocksandbridgesofaWDS,(2)thebridgescanbesolvedbyusingalinear
processsimilartotheforestcomponent,and(3)thenusesanyWDSsolutionmethod,for
exampleGGA,RCTM,orGMPA,to,independently,solveeachblock.
TheBBPAcanalsobeusedtoidentifytheforestcomponentofthenetwork. However,
theuseoftheFCPArequireslessoverheadthantheBBPA.
Thesametopologicalproperties exploitedbyFCPAandBBPAarepartlyresponsible
forthesavingsachievedbypartial-update(AbrahamandStoianov2015). Theforestand
bridgecomponents- beinglinear-convergeafterjustone iterationofapplicationof anon
linear solver. The partial update scheme is able to exploit this by checking for convergence
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Fig. 5.4. The condition number of the Schur complement at the solution for each block
(scatterpoint)andtheconditionnumberoftheSchurcomplementforthefullsystem(red
line)
5.11 Conclusions
In this paper, the bridge-block partitioning algorithm is introduced. The BBPA is a
pre-processing and post-processing algorithm that (1) first partitions the network into
bridge components and block components, (2) then solves for the flows in the bridge
components by a linear process, (3) after that it separately solves for the flows and the
estimated heads for each independent block by using any WDS solver, and (4) finally
the heads are recovered by a linear process at the end. This partitioning of the network
canbeusedtospeed-upthesolutionprocessofthesteadystatedemand-drivenhydraulic
simulationandtoimprovethereliabilityoftheresultsifthecorecomponentoftheWDS
graphisone-connected. Thespeed-upofthesolutionprocessisachievedby(1)solving
the bridge component in the BBPA by a linear process similar to that of solving for the
forest in the FCPA, which reduces the number of non-zeroes in the Schur complement
(2)solvingeachblockbyusingtheminimumnumberofiterationsthatisrequiredbythat
block. Moreover,theBBPAimprovesthereliabilityoftheresultsbecausethecondition
numberoftheSchurComplementforeachblockisboundedabovebytheconditionnumber
fortheSchurComplementofthefullsystem.
The usefulness of the BBPA has also been demonstrated by applying it to eight
benchmark networks with between 934 and 19,647 pipes and between 848 and 17,971
nodes. The total savings in wall clock time after applying the BBPA to the GGA are
between33%and70%. Itisshownthat,thenumberofiterationsandtheconditionnumber
requiredbyeachblockareboundedbythenumberofiterationsandtheconditionnumber
requiredbythefullsystem,respectively. TheuseoftheBBPAcanalsominimizetheneed
to regularize the zero flows when the head loss is modelled by the Hazen-William head
loss equation. This is because in real life systems, such as the case study networks used
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Theproposedframeworksignificantlyreducesthecomputationloadofeachofthesolution
methods that are implemented in WDSLib. This is achieved by categorising each of
the functions that are used in each of the solution methods into three categories: (1)
the functions that will only have to be executed once are called level one (L ) functions.
1
L functions relate to network topology, which is invariant for the whole simulation;
1
(2) in a multi-simulation setting, certain functions will need to be run once for every
hydraulic-phase. These,once-per-assessmentfunctions,arecalledleveltwo(L )functions;
2
and(3) foreveryhydraulicassessment, thereisa non-lineariterativephase inthe solution
process. Thefunctionsinthisphaserunmanytimesforeachhydraulicassessmentuntilthe
stoppingtesthasbeensatisfied. Theseiterative-phasefunctionsarecalledlevelthree(L )
3
functions. Equippedwithsuchaframework,itispossible(1)toconductafaircomparison
betweendifferentsolutionmethods;and(2)toalloweachfunctiontoberuntheminimum
numberoftimesdeterminedbyitssimulationsetting.
Use the proposed framework to conduct a benchmark study on four different WDS
solution methods Theproposedframeworkis thenusedinChapter4to benchmarkthe
performanceoffoursolutionmethods,theglobalgradientalgorithm(GGA),theGGAwith
theforest-corepartitioningalgorithm(FCPA),thereformulatedco-treeflowsmethod,and
theRCTMwiththeFCPA,againsteachother. Eachofthefoursolutionmethodsisapplied
toeightcasestudynetworks.
ProposeanewpartitioningalgorithmtoimprovetheexistingWDSsolutionmethods
In Chapter 5, a new graph partitioning algorithm, bridge-block partitioning algorithm
(BBPA), is proposed. The BBPA is a pre-and-post-processing algorithm that partitions
theWDSgraphintoanumberofbridgecomponentsandanumberofblockcomponents.
EachofthebridgecomponentscanbesolvedusingalinearprocesssimilartotheFCPA
and each of the block components can be separately solved by using any WDS solution
method,theGGA,RCTM,orGMPA.ThereisanumberofadvantagestousingtheBBPA:
(1)thenumberofiterationsrequiredbyeachblockisboundedabovebythatrequiredbythe
unpartitionedsystem–solving theflowsandheadsin eachblock separatelysignificantly
reduces the overall computational time for the non-linear solver in almost all cases; (2)
the condition number of the Schur complement of each block is bounded above by that
of the unpartitioned system. In most cases, the condition numbers for all the individual
blockswillbesmallerthantheconditionnumberofthefullmatrix;(3)thesolutionofeach
blockcanbefoundinparallelinademand-drivenmodelbecausetheflowsandheadsin
oneblockcomponentareindependentfromthoseoftheotherblockcomponents.
6.3 Recommendations for WDS demand-driven solution
methods
The performance of any water distribution system solution method is very problem
dependent. To date, there has been no reliable method that accurately predicts the
performanceofagivenalgorithmonaparticularnetworkapriori. Thisisreflectedinthe
performancedifferencesreportedinthisthesis.
The network topology is the most influential factor in the performance of different
solutionmethods(matrixdensity,thedistributionofnon-zeroelementsafterbandwidth
reduction,etc.). Recommendationsaregivenasfollows:
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SimulationTestBed
real-time monitoring and calibration of the network elements in a supervisory control and
13
data acquisition (SCADA) operational setting, and (4) to adjust control devices, such as
14
valves, in a management setting. In the design setting and both the above operational
15
settings, repeated hydraulic assessment is required on a network with fixed topology. In
16
the management setting, repeated hydraulic assessment is required on a network with
17
flexible network parameter settings. With ever-increasing network sizes and the need for
18
real-timemanagementusingaSCADAsystem, itisimportanttohavearobustsimulation
19
package which can be configured to be maximally efficient whatever the setting.
20
In the field of hydraulic simulation, the system of equations can be formulated as a
21
large and sparse non-linear saddle point problem. There are several well-known iteration
22
methods for solving the non-linear saddle point problem. These include: range space
23
methods (Global Gradient Algorithm (Todini and Pilati 1988)), Null space methods (Co-
24
Tree flow formulation variations (Rahal 1995; Elhay et al. 2014)), loop-based methods
25
(Loop flow correction (Cross 1936)), and pre-and-post-processing methods (forest-core
26
partitioning algorithm (Simpson et al. 2014), domain decomposition (Diao et al. 2014),
27
network clustering (Perelman and Ostfeld 2011)). Their relative performance in terms of
28
speed, rate-of-convergence, and accuracy depends among other things on the topology of
29
the target network: size of the forest component, the number of network loops, and the
30
density of these network loops. It is difficult to evaluate the impact of these topology fac-
31
tors by only examining the incidence matrix that describes the pipe network connectivity.
32
As a result, the best method to use for a particular network cannot be easily determined a
33
priori. Moreover, extra complexity is introduced when a multi-run hydraulic assessment
34
is required. During a multi-run hydraulic simulation, the elapsed computation time of
35
each method can be broken down into two parts: the components that are only required
36
to be performed once at the very beginning for the same network, called the overhead,
37
and the components that are required to be carried out repeatedly for each separate run
38
until the required number of iterations has been met, called the hydraulic-phase. It is
39
desirable to have a simulation platform, given the different levels of repetition, to im-
40
plement these alternative algorithms efficiently. Equipped with such a platform a user
41
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would be able to easily benchmark the performance of alternative methods on a small
42
number of evaluations for a given network and use that performance to inform the choice
43
of algorithm to use for either a once-off simulation setting or for a multiple simulation
44
setting (such as for an evolutionary algorithm (EA)).
45
ThisworkdescribesanextensibleWDSsimulationplatformcalledWDSLib. WDSLib
46
is a numerically robust, efficient and accurate C++ library that implements many WDS
47
simulation methods. WDSLib is written using a modular object-oriented design which
48
allows users to easily mix and interchange solution components, thereby enabling users
49
to avoid redundant computations. It has been optimized to use sparse data structures
50
which are oriented to the pattern of access required for each solution method. WDSLib
51
has been validated for accuracy on a range of realistic benchmark water distribution
52
networks against reference implementations and tested for speed. The program accepts
53
the input file formats of the industry standard EPANET2 (Rossman 2000) toolkit and
54
its performance is faster than EPANET2 in all tested settings and benchmarks.
55
Theremainderofthispaperisstructuredasfollows. Thenextsectiondescribesrelated
56
methodologies and implementations. A general description of the WDS demand-driven
57
steady-state problem is given in the next section. Section 3 presents a mathematical
58
formulation of the network and the solution methods that are used in WDSLib. The
59
tool-kit structure is then given in section 4. This is followed, in section 5, by the toolkit
60
implementation details. Section 6 provides some examples of how the toolkit can be
61
utilized in a simulation work flow. The results are discussed in Section 7. Finally,
62
section 8 summarizes the results of this paper and describes future extensions to the
63
toolkit.
64
2 Background
65
This section describes related water distribution system network solution methods and
66
implementations. The first sub-section describes solution methods, including those used
67
by WDSLib. This is followed by a description of currently available implementations and
68
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compares these with WDSLib.
69
2.1 Related Methods
70
This research considers a water distribution model made up of energy conservation equa-
71
tionsandthedemanddrivenmodelcontinuityequations. TheHardyCrossmethod(Cross
72
1936), also known as the loop flow corrections method, is one of the oldest methods and
73
uses successive approximations, solving for each loop flow correction independently. It
74
is a method that was widely used for its simplicity at the time when it was introduced.
75
More than three decades later, Epp and Fowler (1970) developed a computer version of
76
Cross’s method and replaced the numerical solver with the Newton method, which solves
77
for all loop flow corrections simultaneously. However, this method has not been widely
78
used because of the need (i) to identify the network loops, (ii) to find initial flows that
79
satisfy continuity and (iii) to use pseudo-loops.
80
The GGA is a range space method that solves for both flows and heads. It was the
81
first algorithm, in the field of hydraulics, to exploit the block structure of the Jacobian
82
matrix to reduce the size of the key matrix in the linearization of the Newton method.
83
The GGA has gained popularity through its rapid convergence rate for a wide range
84
of starting values. This is the result of using the Newton method on an optimizations
85
problem that has a quadratic surface. However, it was reported by Elhay and Simpson
86
(2011) that the GGA fails catastrophically in the presence of zero flows in a WDS when
87
the head loss is modeled by the Hazen-Williams formula. Regularization methods have
88
been proposed by both Elhay and Simpson (2011) and Gorev et al. (2012) to deal with
89
zero flows when the head loss is modeled by the Hazen-Williams formula.
90
The GGA as it was first proposed, applied only for the WDSs in which the head loss
91
is modeled by the Hazen-Williams formula, where the resistance factor was independent
92
of flow. Rossman (2000) extended the GGA to allow the use of the Darcy-Weisbach
93
formula. It has been pointed out in Simpson and Elhay (2010), however, that Rossman
94
incorrectlytreatedtheDarcy-Weisbachresistancefactorasindependentoftheflow. They
95
introduced the correct Jacobian matrix to deal with this. It has been demonstrated that
96
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once the correct Jacobian matrix is used, the quadratic convergence rate of the Newton
97
method is restored. Furthermore, Elhay and Simpson (2011) reported that the GGA
98
does not fail in the presence of zero flows when the derivatives of the Darcy-Weisbach
99
Jacobian matrix are correctly computed for laminar flows.
100
The co-trees flow method (CTM) (Rahal 1995) is a null space method that solves for
101
the co-tree flows and spanning tree flows separately. The CTM, unlike the loop flow cor-
102
rections method, does not require the initial flows to satisfy continuity. However, it does
103
require: (i) the identification of the associated circulating graph; (ii) the determination of
104
the demands that are to be carried by tree branches; (iii) finding the associated chain of
105
branches closing a circuit for each co-tree chord; (iv) computing pseudo-link head losses.
106
The reformulated co-trees flow method (RCTM) (Elhay et al. 2014) is also a null space
107
method that solves for co-tree flows and spanning trees flows separately. It represents a
108
significant improvement on the CTM by removing requirements (i) to (iv) above. It uses
109
theSchilders’factorization(Schilders2009)topermutethenode-arcincidencematrixinto
110
an invertible spanning tree block and a co-tree block. This permutation reduces the size
111
of the Jacobian matrix from the number of junctions (as in the GGA) to approximately
112
the number of loops in the network.
113
Abraham and Stoianov (2015) proposed a novel idea to speed-up the solution process
114
when using a null space method to solve a WDS network. Their idea exploits the fact
115
that a significant proportion of run-time is spent computing the head losses. At the
116
same time, flows within some pipes exhibit negligible changes after a few iterations. As a
117
result, there is no point in wasting computer resources to re-compute the pipe head losses
118
for the pipes that have little or no change in flows. This partial update can be used to
119
economize the computational complexity of the GGA, the RCTM and their variations.
120
The forest-core partitioning algorithm (FCPA) (Simpson et al. 2014) speeds up the
121
solution process in the case where the network has a significant forest component. This
122
algorithm permutes the system equations to partition the linear component of the prob-
123
lem, which is the forest of the WDS, from the non-linear component, which is the core
124
of the WDS. It can be viewed as a method that simplifies the problem by solving for
125
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the flows and the heads in the forest just once instead of at every iteration. The FCPA
126
reduces the number of pipes, number of junctions, and the dimension of the Jacobian
127
matrix in the core by the number of forest pipes (or nodes).
128
The graph matrix partitioning algorithm(GMPA) (Deuerlein et al. 2015) exploited
129
the linear relationships between flows of the internal trees within the core and the flows
130
of the corresponding super-links after the forest of the network has been removed. This
131
was a major breakthrough. The GMPA permutes the node-arc incidence matrix in such
132
a way that all of the nodes with degree two in the core can be treated as a group. By
133
partitioning the network this way, the network can be solved by a global step, which
134
solves for the nodes with degree greater than two (super nodes) and the pipes which
135
connect to them (path chords), and a local step, which solves for the nodes with degree
136
two (interior path nodes) and pipes connected to them (path-tree links).
137
2.2 Related Implementations
138
EPANET 2 (Rossman 2000) is a widely used WDS simulation package. EPANET 2 im-
139
plemented the GGA to provide a demand-driven steady-state solution of a WDS. The
140
code for EPANET 2 is in the public domain, allowing many extensions to be devel-
141
oped. Currentlyavailableextensionsinclude: theimplementationofapressure-dependent
142
model (Cheung et al. 2005; Morley and Tricarico 2008; Siew and Tanyimboh 2012; Jun
143
and Guoping 2012) and a real-time simulation capability (Vassiljev and Koppel 2015).
144
The EPANET 2 implementation is not explicitly designed to necessarily be easy to
145
understand or accommodate alternative solution methods (Guidolin et al. 2010). The
146
elements that are used in EPANET 2 are stored by the variables that describe their
147
graph properties. For example, (1) junctions, reservoirs, and tanks are stored as a C
148
struct called Node and (2) all valves, pipes, and pumps are stored as a C struct called
149
Link. The abundant use of global variables limits the reusability and the possibility of
150
the thread-safe design (Guidolin et al. 2010).
151
Consequently, itisdifficulttocleanlyincorporatenewsolutionmethodsintoEPANET
152
2 in a manner that allows a fair comparison of performance between these methods.
153
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Moreover, because there are no clearly defined interfaces for the incorporation of third-
154
party code components in EPANET 2, there is no guarantee that independently authored
155
extensions will be easy to combine with each other.
156
Intheabsenceofapopulareasy-to-modifyWDSsimulationplatformthereiscurrently
157
no straightforward means for comparing different solution methods. To date, when new
158
solution methods have been developed they have been compared using different research
159
systems, on different platforms with different implementation languages. This leads to
160
difficulty in comparing methods, limits the reusability of code, and creates a barrier for
161
researchers to reproduce and replicate results. To address these issues, an extensible
162
framework is required that allows implementation of new methodologies to be easily
163
incorporated without an adverse impact on the performance of the rest of the system.
164
To this end, a number of attempts have been made to implement an object-oriented
165
wrappertoencapsulatetheEPANET2solver(openNet(Morleyetal.2000)andOOTEN(van
166
Zyl et al. 2003)). However, these two systems were focused on providing more flexibility
167
in the processing of input to the core EPANET solver. They did not address any is-
168
sues relating to the solution process. CWSnet, a C++ implementation in object-oriented
169
style, was produced by Guidolin et al. (2010) as an alternative to EPANET 2.0. In CWS-
170
net, more attention has been given to the hydraulic elements of the WDS network. In
171
addition, CWSNet provides a pressure driven model, and takes advantage of the comput-
172
ing power of the computer’s Graphics Processing Unit (GPU). However, in CSWnet the
173
data structures representing the network are specialized to the solution methods that it
174
uses. These data structures are not easily adapted to work efficiently with the different
175
traversal orders, and graph algorithms used by newly developed solution methods. How-
176
ever, CWSnet still uses the same hydraulic solver and the same linear solver techniques
177
implemented in EPANET 2 (Guidolin et al. 2010).
178
To accommodate the deficiencies referred to above, this paper presents a new hy-
179
draulic simulation toolkit WDSlib. WDSlib is coded in C++, and incorporates a number
180
of recently published techniques. This toolkit offers users the ability to: (i) choose from,
181
or modify, different approaches and implementations of different WDS model analyses,
182
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Let q = (q ,q ,....q )T denote the vector of unknown flows, h = (h ,h ,....h )T
209 1 2 np 1 2 nj
denote the vector of unknown heads, r = (r ,r ,....r )T denote the vector of resistance
210 1 2 np
factors, d = (d ,d ,.....d )T denote the vector of nodal demands, e = (e ,e ....e )T
211 1 2 nj l l1 l2 lnf
denote the vector of fixed head elevations.
212
The head loss exponent n is assumed to be dependent only on the head loss model:
213
n = 2 for the Darcy-Weisbach head loss model and n = 1.852 for Hazen-Williams head
214
loss model. The head loss within the pipe j, which connects the node i and the node k,
215
216
is modelled by h
i
h
k
= r jq
j
q
j
n −1. Denote by G(q) Rnp ×np, a diagonal square matrix
− | | ∈
217
with element [G]
jj
= r
j
q
j
n −1 for j = 1,2,....n p. Denote by F(q) Rnp ×np, a diagonal
| | ∈
square matrix where the j-th element on its diagonal [F] = d [G] q . A is the full
218 jj dqj jj j 1
rank, unknown head, node-arc incidence matrix, where [A ] is used to represent the
219 1 ji
relationship between pipe j and node i; [A ] = 1 if pipe j enters node i, [A ] = 1 if
220 1 ji 1 ji
−
pipe j leaves node i, and [A ] = 0 if pipe j is not connected to node i. A is the
221 1 ji 2
fixed-head node-arc incidence matrix, where [A ] is used to represent the relationship
222 2 ji
between pipe j and fixed head node i, [A ] = 1 if pipe j enters fixed head node i,
223 2 ji
−
[A ] = 1 if pipe j leaves fixed head node i, and [A ] = 0 if pipe j is not connected to
224 2 ji 2 ji
fixed head node i.
225
3.2 System of Equations
226
The steady-state flows and heads in the WDS system are modeled by the demand-driven
227
model (DDM) continuity equations (1) and the energy conservation equations (2):
228
A Tq d = O (1)
1
− −
229
G(q)q A h A e = O, (2)
1 2 l
− −
which can be expressed as
230
G(q) A q A e
1 2 l
− = 0, (3)
−
A T O h d
1
−
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as the generalized equations that can be applied when the head-loss is modeled by the
245
Hazen-Williams equation or the Darcy-Weisbach equation. The correct Jacobian matrix
246
with the formula for F, when head loss is modeled by Darcy-Weisbach equation, can be
247
found in Simpson and Elhay (2010). They showed that the use of the correct Jacobian
248
matrix restores the quadratic rate of convergence.
249
It is important to note that the GGA, as it was originally proposed, solves the entire
250
networkbyanon-linearsolver,andthiscanincludesomeunnecessarycomputationswhich
251
can be avoided by exploiting the structural properties of the WDS graph composition.
252
The methods described below exploit these structural properties to potentially improve
253
the speed of the solution process.
254
3.4 Forest-Core Partitioning
255
Associated with a WDS is a graph G = (V,E), where the elements of V are the nodes
256
(vertices) of the graph G and elements of E are the pipes (links) of the graph G. The
257
graph G can be partitioned into smaller subgraphs with special properties. The special
258
properties that are exploited in WDSLib and their formulations are described in this
259
subsection. The concept of partitioning the WDS network was proposed by Deuerlein
260
(2008) in order to simplify the WDS solution process. Simpson et al. (2014) extended
261
the idea of the network partitioning of Deuerlein (2008) and introduced the forest-core
262
partitioning algorithm (FCPA), which partitions the network into a treed component
263
and a looped or core component. The FCPA starts with a searching algorithm which
264
265
identifies the forest subgraph, G
f
= (V f,E f), in which S Nnf×np is the permutation
∈
matrix which identifies the pipes in the forest, E , as distinct from the pipes in the
266 f
267
core , E c, and T Nnf×nj is the permutation matrix which identifies the nodes in the
∈
forest, V , as distinct from the nodes in the core, V , as distinct from the core subgraph,
268 f c
269
G
c
= (V c,E c), in which P Nnpc×np is the permutation matrix for E
c
and C Nnjc×nj
∈ ∈
is the permutation matrix for V .
270 c
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The flows of the pipes in the forest, Sq, can be found directly from
271
Sq = TA TST −1 Td. (10)
272 1
−
(cid:0) (cid:1)
The system for the reduced non-linear problem (for the core heads and flows) can be
273
expressed as
274
PGPT PA CT Pq PA e
1 2 l
− = , (11)
275
CA TP O Ch Cd+CA TSTSq
1 1
−
and then the Newton iterative method is applied to Eq. (11).
276
Finally, once the iterative solution process for the core has stopped, the forest heads
277
can be found by solving a linear system:
278
Th = SA TT −1 SA e SGSTSq +SA CTCh . (12)
279 1 2 l 1
− −
(cid:0) (cid:1) (cid:0) (cid:1)
The system for the reduced non-linear problem (for the core heads and flows) in Eq. 11
280
can be expressed as:
281
G A q A e
1 2 l
− = (13)
282
A T O h d
− b1 b b b
where G = PGPT,A = PAb CT,q = Pqb,h = Chb,A = PA , and d = Cd +
283 1 1 2 2
CA TSTSq.
284 1 b b b b b b
The FCPA simplifies the problem by identifying the linear part of the problem and
285
solving it separately from the core to avoid unnecessary computation in the iterative
286
process.
287
3.5 Reformulated Co-Tree flows method
288
A graph, with or without forest, can be partitioned into two sub-graphs: a spanning
289
tree subgraph and a complementary co-tree subgraph. The reformulated co-tree flow
290
method (RCTM) (Elhay et al. 2014) exploited the relationship between the spanning tree
291
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nent of the solver to n n (the number of co-tree elements in the network). It has been
315 p j
−
proven by Elhay et al. (2014) that the RCTM and the GGA have identical iterative re-
316
sults and solutions if the same starting values are used. However, for the RCTM, the user
317
(0)
only needs to set the initial flow estimates for the co-tree pipes, q , in contrast to GGA
318 2
where initial flow estimates are required for all pipes. The flows in the complementary
319
spanning pipes are generated by Eq.(14) in the RCTM.
320
4 WDSLib Structure
321
WDSLib is a WDS simulation toolkit consisting of a set of C++ member functions,
322
which henceforth will be referred to just as functions, that can be composed to solve
323
for the steady state solution of a WDS. WDSLib can be used for a once-off simulation
324
or a multi-run simulation. Pre-packaged driver code is provided to perform once-off
325
simulations using a choice of solver methods. For a multi-simulation setting, where the
326
use-cases are very diverse, the user is able to select the desired components of WDSLib
327
to compose and compile their own driver.
328
Individual functions in WDSLib are classified according to their role in the simulation
329
workflow. In any simulation workflow, there will be functions that will only have to be
330
executed once. For example, functions to read the input file or partition the network will
331
only have to execute once at the start of the simulation (or of all simulations). Likewise,
332
code to reverse the network partitioning and write simulation results will only have to
333
execute once at the end of the simulation. In this work, these functions that are only
334
required to be run once are called level one (L ) functions. L functions relate to network
335 1 1
topology, which is invariant for the whole simulation. In a multi-simulation setting,
336
certain functions will need to be run once for every hydraulic-phase. An example of
337
such a module is the module making the initial guesses of pipe flow rates for the updated
338
network configuration. In this work, these, once-per-assessment functions, are called level
339
two (L ) functions.
340 2
Finally, for every hydraulic assessment there is a non-linear iterative phase in the so-
341
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lution process. The functions in this phase run many times for each hydraulic assessment
342
until the stopping test has been satisfied. Examples of these include the functions to
343
calculate the G and F matrices (see Eqs. (3) and (4)) and running the Cholesky solver.
344
These iterative-phase functions are called level three (L ) functions.
345 3
Fig. 1 illustrates the global structure of WDSLib under a once-off simulation setting
346
and a multi-run simulation setting. The modular setup of WDSLib allows each module
347
to be run the minimum number of times determined by its simulation setting. Under
348
the module structure described above a once-off simulation setting can be viewed as a
349
special case where the L functions and L functions are both run once. Note that after
350 1 2
running the initial L functions it is possible to run hydraulic assessments of the network
351 1
in parallel. This mode of execution might be used in a design setting such as using a
352
genetic algorithm (GA) to optimize pipe diameter sizes.
353
Figure 1: Global structure of WDSLib for both simulation settings
L and L functions are classified into parts a and b according to whether they run
354 1 2
before or after the lower level processing that they embed. These functions are detailed
355
in Fig. 2. The L functions that run at the start of the simulation are called L func-
356 1 1a
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tions. These include the module to read the configuration file and the EPANET .inp file;
357
partition the network; and solve the linear part(s) of the network. The corresponding
358
L functions are run at the end of the simulation. These include tasks such as reversing
359 1b
the network partitioning. Note that certain L functions require their corresponding L
360 1a 1b
functions to be used. For example the forest search module needs to be paired with the
361
reverse FCPA permutation. There is a similar structure for L functions. L functions
362 2 2a
are run at the start of each hydraulic assessment and L functions run at the end. The
363 2b
functions that must be included for the FCPA method are denoted with single asterisks.
364
Likewise the functions that must be included with the RCTM method are denoted with
365
double asterisks. For these methods to work correctly all affiliated functions must be
366
included in the simulation workflow. Note that it is also possible to run both the RCTM
367
and FCPA in the same workflow. Also note that the user cannot run both GGA and
368
RCTM in the same workflow – the user must choose between these solution methods.
369
Table 1 provides a mapping from the function descriptions in Fig. 2 to the function
370
names in WDSLib. In addition, the dependencies between functions for each solution
371
method are shown in Table 1a, Table 1b, Table 1c and Table 1d. The columns in each
372
table list, respectively, the description of the function, its name in WDSLib, the C++
373
class in which it appears, its input parameters, and its output values. Note, that void
374
is used in these latter two columns to denote that the function interacts with the class
375
variables rather than through its parameters and return value. Examples of how these
376
functions can be coded are presented in section 6. The key data-access functions in
377
WDSLib are described next.
378
Getter and Setter methods Each class in WDSLib has various methods available for
379
setting the network parameters and retrieving the results of the WDS network. These
380
methods allow the user to reconfigure the network before and during simulation runs.
381
The names of the setter methods all start with a prefix set and the names of the getter
382
methods all start with a prefix get. For example, a user can set (write-to) the diameter
383
of pipe index to value by calling pipe->setD(index,value) and get (read-from) the
384
head of node index by calling h[index]=result->gethFinal(index). A summary of
385
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Table 1: Key function descriptions, names, their classes, inputs and outputs. The affili-
ated functions are shown in sub-tables (1a) (1b) (1c) (1d).
(a) Shared Modules
Description Modulename Class Input Output
Readtheconfigurationfile readConfig runManager configfilename void
ReadEPANETinputfile getInputData Input EPANET.inpfile EPANETerrcode
Variablesscaling scale Solver void void
AMDbandwidthreduction AMD Suitesparse void void
Calculatetheresistanceconstants getRf Solver net resistanceconstant
Generateinitialguessesofflows init Solver diameter flowrate
Calculatetheheadlosscoefficients getGF Solver net,resistanceconstant void
Stoppingtest stopTest Solver result norm
Recoverscaledvariables rScale Solver void void
(b) Global gradient algorithm (GGA)
Description Module name Class Input Output
GGA Solver runH GGASolver(Solver) void void
(c) Forest-core algorithm (FCPA)
Description Module name Class Input Output
Forest search forestSearch topology SN, EN void
Calculate flows in forest forestFlow solver demands flows in forest pipes
Calculate heads in forest forestHead solv result heads in forest pipes
Reverse FCPA permutation rFCPA Solver void void
(d) Reformuated cotree flows method (RCTM)
Description Module name Class Input Output
Spanning tree search STSearch topology SN, EN void
RCTM solver runH RCTMsolver (Solver) void void
Calculate heads in ST and CT RCTMHead RCTMsolver flows in ST and CT void
Reverse RCTM permutation rRCTM RCTMsolver void void
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Table 2: The getter and setter functions of each class and the variables they access
Class Name Description Read-Access Write-Access
Net Basic network properties, & Pipe and Node Node,Pipe, n , n ,
p j
n
s
Node Node properties d, z , z
s u
Pipe Pipe properties SN, EN, L, D, R, pipe ID
Flag Flag information getFlag(”flagN”,flagV) setFlag(”flagN”)
Parameter Parameter information getPara(”paraN”,paraV)setPara(”paraN”)
Simulation Manage hydraulic simulation - -
Solver Parent class of solution methods - -
GGASolver GGA solution method Result -
RCTMSolver RCTM solution method Result -
Topology Network topology information getCore, getForest
Result Results of the simulations qIter, hIter, GIter ,
FIter, numIter, Cre-
sIter, EresIter, Time
the variables that can be read-from (read-access through getter methods) and written-to
386
(write-access through setter methods for each key classes is specified in Table 2. This
387
concludes the discussion of the the broad structure of the WDSLib package. The next
388
section describes key aspects of the implementation of the package.
389
5 WDSLIB: Toolkit Implementation
390
This section outlines key implementation details of WDSLib. As previously mentioned,
391
the overall aim of WDSLib is to provide a clearly-structured, flexible and extensible hy-
392
draulic simulation toolkit that allows testing, evaluation, and use, in production settings,
393
of both existing and new WDS solution techniques. These aims require WDSLib be im-
394
plemented so that it is fast to execute, flexible to configure, robust to challenging input
395
datacases, andeasytounderstandandmodify. Thefollowingdescribesaspectsoftheim-
396
plementation of WDSLib that enable it to meet these requirements. The next subsection
397
describes the general considerations that informed the design of the whole toolkit. This
398
generaldiscussionisfollowedbyasummaryofkeyimprovementstothesolutionprocesses
399
encoded in forest searching and spanning tree searching in the WDSLib package.
400
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5.1 General capabilities and properties
401
This sub-section describes design aspects underpinning the utility and performance of
402
WDSLib. In-turn, the following outlines measures taken to: (1) maximize code clarity
403
and modularity; (2) increase the efficiency of memory access and storage; (3) maximize
404
numerical robustness; (4) facilitate accurate timing of code execution; and (5) maximize
405
simulation speed for different settings.
406
Design Considerations 1: Modularity
407
The modular design of WDSLib is central to the evaluation and testing of different WDS
408
solution methods. All methods have been defined to perform a single, well–defined,
409
function and each class can be compiled, used and tested independently. These features
410
allowuserstoassemblethemethodsofinterestfromindependentlydevelopedcomponents
411
tocreateacustomizedWDSsolutionmethodinareliableway. WDSLib’smodulardesign
412
also allows the users to profile the computation time of each individual component of an
413
algorithm. Functions communicate through well-defined interfaces and the function code
414
has been factored to minimize development and testing cost. This architecture allows
415
customized simulation applications (i) to combine the functions of interest and (ii) to
416
implement new solution algorithms to extend the functionalities of WDSLib.
417
Design Considerations 2: Memory Considerations
418
Care was taken to minimize the memory footprint of executing code (in order to re-
419
duce memory requirements and prevent memory leaks) in the interest of the toolkit
420
efficiency and toolkit robustness. Reducing memory requirements allows the solution of
421
larger WDS problems for a given memory capacity. In WDSLib, memory reduction was
422
achieved through both, using sparse matrix representations and the systematic allocation
423
and deallocation of working structures in the C++ code. The matrices used in WDS
424
simulation are often sparse, with the density of the full node-arc incidence matrix being
425
only 2/n . Consequently, it is more efficient to store these matrices using sparse stor-
426 j
age schemes which store only the non-zero elements of the matrix and pointers to their
427
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SimulationTestBed
locations (Davis et al. 2014). It is important to note that the choice of a sparse ma-
428
trix representation is made based on (1) the storage requirements of the matrix and (2)
429
common search orders to column elements and row elements. This latter factor means
430
that the best format for sparse matrix representation varies with the preponderant or-
431
ders of search, (row-wise, column-wise, or both), employed by each method. There is
432
a number of common storage formats for sparse matrices (Compressed column storage
433
(CCS) of Duff et al. (1989)), Compressed row storage (CRS), Block Compressed column
434
storage (BCCS), Block Compressed row storage (BCRS), and Adjacency lists). As will
435
be described shortly, WDSLib, uses a modified adjacency-list representation.
436
Other implementations use a variety of storage schemes. In EPANET 2, the A
437 1
matrix is stored as two arrays of node indices, which represent start nodes (SN) and the
438
end nodes (EN) of each pipe. The i th entry of the SN and EN arrays represent the
439
−
start node and end node of i th pipe of the network. This storage format minimizes
440
−
the memory required to store the A matrix because only the indices are required to
441 1
be stored because [A ] = 1 and [A ] = 1. As shown in Table 4, searching
442 1 (i,SNi)
−
1 (i,ENi)
through rows (pipes) of matrices that are stored in this format is efficient. However,
443
searching though the columns (nodes) is relatively inefficient. This storage format is also
444
used in CWSnet.
445
Both CCS and CRS are used in the FCPA implementation reported in Simpson
446
et al. (2014), and the RCTM implementation reported in Elhay et al. (2014). The
447
partial update null space method (Abraham and Stoianov 2015) used CCS. The memory
448
requirement for storing the A matrix in CCS is 2 nnz +n +1 as shown in Table 4.
449 1 j
×
This storage scheme is fast for searching through columns (nodes) of matrices that are
450
stored in CCS and slow for searching though rows (pipes).
451
In WDSLib, a modified adjacency list, described in Table 3, tailored for WDS hy-
452
draulic simulation, is used. An adjacency list for an undirected and unweighted graph
453
consists of n unordered lists for each vertex n , which contains all the vertices to which
454 j i
vertex n is adjacent. The network that is shown in the Fig. 3 has one source, three
455 i
nodes, and four pipes. The adjacency list for this network can be described by four lists
456
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SimulationTestBed
Table 3: The adjacency-list matrix presentation
Node Index adjacent to Size
1 (v ,e ) v N(v ) e connects v and v Deg(v )
i j i 1 j 1 i 1
{ | ∈ }
2 (v ,e ) v N(v ) e connects v and v Deg(v )
i j i 2 j 2 i 2
{ | ∈ }
3 (v ,e ) v N(v ) e connects v and v Deg(v )
i j i 3 j 3 i 3
. { | ∈ . } .
. . .
. . .
n (v ,e ) v N(v ) e connects v and v Deg(v )
j
{
i j
|
i
∈
nj j nj i
}
nj
2,3 , 1,4 , 1,4 , 2,3 . Each list describes the set of adjacent vertices of a vertex
457
{{ } { } { } { }}
in the graph. For example, the first list, 2,3 , represents that the vertex 1 is adjacent
458
{ }
to the vertex 2 and vertex 3.
459
Figure 3: A simple sample network. Numbers denote junction and pipe indices in the
network.
The adjacency list is modified to include a directed and weighted graph for WDSLib.
460
This modified adjacency list for a directed and weighted WDS graph consists of n un-
461 j
ordered lists for each vertex n . This list contains all the vertex and edge pairs to which
462 i
vertex n is adjacent. For example, the adjacency list for the same network that is shown
463 i
intheFig.3canbedescribedbyfourlists (2,1),(3,4) , (1,1),(4,2) , ((1,4),(4,3) , (2,2),(3,3) .
464
{{ } { } { } { }}
Each list represents the set of adjacent vertex and edge pair of a vertex in the graph. For
465
example, the first list, (2,1),(3,4) , describes that the vertex 1 is adjacent to the vertex
466
{ }
2 by edge 1 and the vertex 3 by edge 4. It is fast to search through both the rows and
467
columns of the A matrices that are stored in this format.
468 1
In addition to these optimized encodings, both G and F are diagonal square matrices,
469
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SimulationTestBed
Table 4: Different sparse representations for A
1
Types size(A ) size(A ) size([A A ]) Column Search Row Search
1 2 1 2
CCS 2 nnz+n +1 2 nnz+n +1 4 n +n +2 O(n) O((n )n)
j f p n j
× × ×
CRS 2 nnz+n +1 2 nnz+n +1 6 n +2 O((n )n) O(n)
p p p p
× × ×
EPANET - - 2 n O(n) O((n )n)
p j
×
WDSlib - - 4 n O(n) O(n)
p
×
which require less storage when stored as vectors than in sparse matrix form. The storage
470
methods used for the variables in WDSLib and their associated memory usage are given
471
in Table 5.
472
As a final note, to offer further assurance of the correctness of memory management in
473
WDSLib, Valgrind (Nethercote and Seward 2007), a programming debugging tool, was
474
deployed during testing to detect any memory leaks, memory corruption, and double-
475
freeing.
476
Table 5: Vectors and matrices in WDSLib
variables type size storage method memory requirements
q, L D, r vector n 1 vector n double
, p p
× ×
h, d vector n 1 vector n double
j j
× ×
G, F matrix n n vector n double
p p p
× ×
A , A matrix n n sparse matrix (2 n ) integer
1 2 p j p
× × ×
L matrix (n n ) n sparse matrix (n n ) n integer
21 p j j p j j
− × ≤ − × ×
Design Considerations 3: Numerical Considerations
477
The calculations in WDSLib are performed in C++ under IEEE-standard double pre-
478
cision floating point arithmetic with machine epsilon (cid:15) = 2.22 10 16. Invariant
479 mach −
×
terms and parameters in every equation were evaluated in advance and replaced by full
480
20-decimal digit accuracy constants. Intermediate results of calculations, (which are not
481
easily accessible in EPANET), can be output at the user’s request. The stopping toler-
482
ance and stopping test can be set by the user either through the configuration file or by
483
the relevant setter method in the Parameter class.
484
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is identified as a leaf node when its node degree is one. Every time a leaf node, node k,
533
is identified, the node pointer is moved to its adjacent node, node k, and the node degree
534
of node k is reduced by one. This process repeats if the adjusted node degree of node k
535
is one. Otherwise, node k is the root node for this tree and the algorithm progresses to
536
the next tree in the forest.
537
Key Optimization 2: Improvements to Spanning Tree Search
538
The reformulated co-tree flows method (RCTM) in this paper is also a substantial im-
539
provement over the algorithm of the original paper (Elhay et al. 2014). The original
540
spanning tree search algorithm sweeps the rows of the A matrix (pipes) in order to
541 1
identify the singleton rows and their corresponding columns. The spanning tree search
542
in the original RCTM required a sweep of of the A matrix to identify the next pipe in
543 1
the spanning tree. This algorithm is O(n n ), which is relatively inefficient.
544 p j
The pseudo-code for the refined spanning tree search algorithm is shown in Ap-
545
pendixCThisimprovedalgorithmtakesasinputtheadjacencylistdescribingthenetwork
546
and the pipe indexes of the core component of the network from the Algorithm 1 (if the
547
FCPA is used). In this algorithm, all water sources are the starting point of the search
548
process, SN, and marked as visited. The nodes in SN are then used as to identify a
549
spanning tree within the WDS. This is achieved by repeatedly finding all adjacent pairs,
550
node t and pipe s, of and removing the first node in SN by using the adjacency list.
551
If the adjacent node t is not visited then node t is inserted into the spanning-tree node
552
vector, STN, and search node vector, SN, and node t is marked as visited and pipe s to
553
the spanning-tree pipe vector, STP, and pipe s is marked as visited. If the adjacent node
554
t is visited and the pipe s is not visited then the pipe s is inserted into the co-tree pipe
555
vector, CTP and mark pipe s as visited. This process is repeated until SN is empty.
556
The overall time-complexity of this algorithm is O(n + n ) (compared to O(n n ) as
557 p j p j
mentioned above) is the same as the best asymptotic complexity of breadth-first search
558
on a graph.
559
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SimulationTestBed
6.1 Example 2 - A Simple Network Design Application
578
AsaminimalistexampleoftheapplicationofWDSLibtoaWDSnetworkdesignproblem,
579
the following example uses 1+1EA for optimally sizing pipe diameters. This algorithm
580
takes an existing network with randomly generated pipe diameters and optimizes the
581
network to minimize cost, subject to given pressure head constraints. A 1+1EA is a very
582
simple evolutionary strategy (Beyer and Schwefel 2002) which starts with a randomly
583
generated individual (in this case a WDS diameter configuration). This 1+1EA then
584
progresses by applying a mutation to a random pipe diameter size, and then evaluating
585
the new individual. If the new individual is better it replaces the old network. This
586
process continues in a loop until a given number of evaluations is reached.
587
The C++ code for this example is shown in Figs. 5, 6, 7, and 8. If the name of the
588
file containing this code is: simpEA.cc then the simplest command to compile this code
589
is:
590
g++ simpEA.cc -o simpEA -Llib -lWDSLib
591
To run this code the user would type:
592
./simpEA config.txt
593
where config.txt contains the same configuration text as for the previous example.
594
Starting with the main function in Fig. 5, line 15 points to the config file specified by
595
thecommandline. Thenexttwolinesinitializetheresultandthesimulationaccording
596
to the configuration file. This is followed by the L module to perform the user selected
597 1a
L functions. Line 19 generates the initial pipe diameters of the network and line 20
598 1a
initializes the workspace for the mutated string. Line 23 sets the pipe diameters of the
599
network. Line 24 evaluates the current network configuration. The permutation and
600
scaling for the current individual is reversed by L in line 25 of Fig. 5. Line 26 calculates
601 1b
the fitness of the current network configuration by using the evaluate function in Fig. 8.
602
This function applies a penalty for pressure head constraint violations and pipe material
603
costs. The body of the 1+1EA is contained in the selection operator and mutation
604
operator that follow. Lines 27 to 31 compare the string in the current generation with
605
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SimulationTestBed
the current best string if the individual p1, as measured by evaluate is better than the
606
individual p2 then p1 replaces p2. Line 32 mutates the current network, p2, using mutate
607
(see Fig. 7). The mutate function changes the diameter of a randomly selected pipe in the
608
networktoarandomlyselecteddiameter, chosenfromasetofcommerciallyavailablepipe
609
diameters. The mutated individual, stored in the workplace p1, is used as the network
610
configuration for the next iteration. Until the total number of generations is reached, the
611
user selected information about the best individual is outputted by dispResult in line
612
34 of Fig. 5.
613
It should be noted that the algorithm described above can be used to design a simple
614
WDS but is not optimal in terms of speed of convergence. Other EA’s such as genetic
615
algorithms (Simpson et al. 1994) will perform better. However the above example has the
616
advantage of simplicity and contains all the basic elements that a GA would use when
617
interacting with WDSLib.
618
This concludes the presentation of examples in this work. The next section presents
619
a case study that illustrates the performance of WDSLib in a multi-simulation setting.
620
7 Case Study
621
The following presents timing results for WDSLib running the 1+1EA described in the
622
previous section. The results below compare the four different solvers plus EPANET2.
623
Note, that detailed timings for once-off simulations comparing the four methods can be
624
found in Qiu et al. (2018). Three networks were benchmarked in these experiments.
625
These were the N , N , and N case-study networks used in Simpson et al. (2014).
626 1 3 4
Table 7 summarizes the characteristics of these networks.
Table 7: Benchmark networks summary
Full Network Forest & Core Networks Co-tree Network
Network n n n n (n /n#) n n n
p j s f f p jc pc ct
N 934 848 8 361 (38%) 573 487 84
1
N 1975 1770 4 823 (42%) 1152 947 205
3
N 2465 1890 3 429 (17%) 2036 1461 757
4
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SimulationTestBed
Table 8 shows the results of the 1+1EA from Fig. 5 for the GGA, GGA with FCPA,
627
RCTM, RCTM with FCPA and the EPANET2 solvers. For each of the four WDSLib
628
solvers above, the timings are given for running the EA with and without the L1 modules
629
hoisted out the main EA loop. Each experiment evaluates the WDS network 100,000
630
times. And the best performing method for each network is highlighted in bold. It is
631
important to note that 1+1EA using both the GGA and the WDSLib
632
Table 8: The actual 1+1EA run-time with 100,000 evaluations (min.) for each of the
four solution methods applied networks N , N , N
1 3 4
GGA GGA with FCPA RCTM RCTM with FCPA EPANET
min. min. min. min. min.
N 6.73 4.64 4.53 4.13 9.81
1
N 15.21 9.79 13.75 10.30 26.43
3
N 21.14 16.29 23.92 21.93 67.11
4
The results show that the EA runs using WDSLib are substantially faster than the
633
runs using the EPANET2 solver. This is, in part, due to the fact that the EPANET2
634
solver is designed as a standalone solver which does not facilitate lifting out of invariant
635
computations from the EA loop.
636
As a demonstration of how the performance of an EA can be traced Fig. 9 shows the
637
evolution of the fitness values of the N network. These traces were extracted from a file
638 1
written to in line 30 in Fig. 5. As can be seen, the cost and the pressure head violation
639
terms drop during the EA run. Note that there will be considerable variation between
640
1+1EA runs due to its highly stochastic nature.
641
8 Conclusions
642
ThispaperhasdescribedWDSLib, alibraryforsteady-statehydraulicsimulationofWDS
643
networks. WDSLib is fast, modular, and portable with implementation of several stan-
644
dard and recently published hydraulic solution methods. We have outlined the supported
645
solution methodologies, the structure of the package and key aspects of WDSLib’s imple-
646
mentation. Two example applications have been presented including a design case study
647
150 |
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