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ADE | reducing the complexity of the problem. According to the results of the foregoing analyses,
the parameters of the uniaxial compressive strength (𝑈𝐶𝑆), Young’s modulus (𝐸) and
horizontal pressure coefficient (𝐾) were identified as the most influential parameters for
modelling of 𝜎 ; while, the parameters of Young’s modulus (𝐸), Poisson’s ratio (𝜐),
𝑅𝐵
horizontal pressure coefficient (𝐾) and 𝜎 were recognised as the best combination of
𝑅𝐵
inputs for modelling of 𝐼 . [see Chapter 5]
𝑅𝐵
• The mathematical functions and the visual patterns provided by the GEP and classification
and regression tree (CART) techniques unravelled the latent relationship between the
rockburst parameters (i.e. 𝜎 and 𝐼 ) and their corresponding influential parameters. The
𝑅𝐵 𝑅𝐵
performance analysis of the developed models showed that the GEP-based models with
the values of 0.94, 14.25 and 9.80 for the indices of 𝑅2, 𝑅𝑀𝑆𝐸 and 𝑀𝐴𝐸 for 𝜎 and the
𝑅𝐵
values of 0.94, 0.19 and 0.14 for the foregoing performance indices for 𝐼 outperformed
𝑅𝐵
the CART-based models. However, the CART algorithm was recognised as the efficient
tool for solving the high-complex non-linear problems in mines. [see Chapter 5]
• The performed parametric analysis on the best models showed that by an increase in 𝑈𝐶𝑆
and 𝜎 , 𝜎 increases monotonically. Also, the risk of rockburst occurrence showed a
𝑣 𝑅𝐵
downward non-linear trend with the independent parameters of 𝐾, 𝐸, 𝜐 and 𝜎 .
𝑅𝐵
Furthermore, the parametric analysis showed strong correlations among the rockburst
parameters and their input parameters, representing that the selected inputs are potential
indicators for assessing and predicting the rockburst phenomenon in deep underground
mines. [see Chapter 5]
• The developed “Double-Criteria Damage-Controlled Test Method” in this research project
by adapting two controlling criteria, including the maximum axial stress level that can be
achieved and the maximum lateral stain amplitude that the rock specimen can experience
in a cycle during loading, i.e. 𝐴𝑚𝑝.(𝜀 ), was successful in capturing the post-peak stress-
𝐿
strain behaviour of Tuffeau limestone subjected to the uniaxial multi-level systematic
cyclic loading history. This technique opens new insights into the rock failure mechanism
and the long-term stability assessment of the underground structures under seismic
disturbances. [see Chapter 6]
• It was found that the overall post-peak behaviour of rocks under multi-level systematic
cyclic loading is characterised by the combination of class I and class II; however, the class
II behaviour was more dominant for the specimens that experienced more loading and
unloading cycles. [see Chapter 6]
277 |
ADE | • The specimens which experienced more cycles in the pre-peak regime failed at stress levels
higher than the determined average 𝑈𝐶𝑆 for Tuffeau limestone specimens, i.e. the strength
hardening occurred. [see Chapter 6]
• The following four main stages were distinguished for the evolution of damage parameters
of Tuffeau limestone specimens subjected to the multi-level systematic cyclic loading
history: (a) The increase in rock stiffness due to the closure of the pre-existing voids and
rock compaction, which was accompanied by the decrease in energy dissipation ; (b) the
domination of the quasi-elastic behaviour due to the balance between two mechanisms of
grain-crushing and pore collapse over the pre-peak domain; (c) the gradual decrease in
rock stiffness due to dilatant microcracking with more energy dissipation; and (d) the
generation and coalescence of microcracks which resulted in a rapid increase in damage
and energy dissipation and more reduction in stiffness. [see Chapter 6]
• According to the evolution of the crack damage stress (𝜎 ) during the cyclic loading, the
𝑐𝑑
rock specimens did not switch from the compaction-dominated to a dilatancy-dominated
state, should the applied stress level is not high enough to create critical damage within the
specimens. This resulted in a constant 𝜎 that is approximately equal to the unloading
𝑐𝑑
stress in each cycle. [see Chapter 6]
• The observed strength hardening for the Tuffeau limestone specimens can be attributed to
the rock compaction induced by the cyclic loading history. The weak bonding between the
grains can be broken during loading and unloading cycles and the produced fine materials
may fill up the internal pores, which finally may result in more rock compaction and
hardening behaviour. [see Chapter 6]
• The proposed testing methodology was also successful in capturing the complete stress-
strain curves (i.e. the pre-peak and the post-peak regimes) of Gosford sandstone
specimens subjected to single-level systematic cyclic loading at different stress levels (i.e.
𝜎 /𝜎 =80-96%). [see Chapter 7]
𝑎 𝑚
• A threshold of 𝜎 /𝜎 was identified which lies between 86-87.5%. For the stress levels
𝑎 𝑚
lower than this range (i.e. the hardening cyclic loading tests), failure did not occur for a
large number of cycles, and the rock specimens followed a two-stage damage evolution
law (dominated by the quasi-elastic behaviour). For these tests, the damage evolution also
was found to be independent of the cycle number, as no considerable effect was observed
on damage parameters by increasing the cycle number from 1500 to 10000 cycles. [see
Chapter 7]
278 |
ADE | • Below the fatigue threshold stress, the rock behaviour under cyclic loading in the pre-peak
and the post-peak regimes was approximately similar to those in monotonic loading
conditions. For the specimens subjected to the cyclic loading below the fatigue threshold
stress, no considerable damage was incurred within the specimens and the peak strength
increased up to 8% after applying the monotonic loading (i.e. the strength hardening
occurred). [see Chapter 7]
• For the specimens which experienced cyclic loading beyond the fatigue threshold stress
(i.e. the fatigue cyclic loading tests), the failure occurred during loading and unloading
cycles. For such tests, the lateral and volumetric irreversible strain were accumulated more
rapidly in the specimens. Moreover, beyond the fatigue threshold stress, the increase in
𝜎 /𝜎 resulted in rock failure in a more brittle/self-sustaining manner. [see Chapter 7]
𝑎 𝑚
𝑖𝑟𝑟
• According to the evolution of the cumulative irreversible axial strain (∑𝜀 ), a secondary
𝑎
inverted S-shaped damage behaviour was identified in the post-peak regime of the fatigue
cyclic loading tests. In other words, the second loose behaviour before the failure point
extends to the post-peak stage for several cycles. These loose hysteretic loops are followed
by a dense behaviour for a large number of cycles until the complete failure of the
specimens occurs, showing another loose behaviour. With the increase of the applied
stress level, the damage per cycle decreased exponentially, and the three stages of the
secondary inverted S-shaped damage behaviour was more visible in the post-peak regime.
[see Chapter 7]
• The modified triaxial testing procedure, i.e. mounting four lateral strain gauges at the mid-
length of the rubber membrane and connecting them to a Wheatstone bridge to provide a
single lateral strain feedback signal, was successful in controlling the axial load and
performing the single-level systematic cyclic loading tests at different stress levels and
confining pressures. [see Chapter 8]
• By increasing the confinement level (𝜎 /𝑈𝐶𝑆 ) from 10% to 100%, the fatigue
3 𝑎𝑣𝑔
threshold stress (FTS) of Gosford sandstone decreased from 97% to 80%, which indicated
that rocks in great depth experience the failure due to cyclic loading at stress levels much
lower than the determined monotonic strength. [see Chapter 8]
• An unconventional post-peak stress-strain behaviour was observed for rocks by an
increase in confinement level (𝜎 /𝑈𝐶𝑆 ) so that for lower 𝜎 /𝑈𝐶𝑆 , the rock
3 𝑎𝑣𝑔 3 𝑎𝑣𝑔
specimens mostly showed a class II/self-sustaining behaviour, while for higher
confinements, the ductile behaviour was dominant. [see Chapter 8]
279 |
ADE | • According to the calculated energy-based brittleness index for the rock specimens which
did not fail in cycles, a transition point at 𝜎 /𝑈𝐶𝑆 = 65% was identified, where the rock
3 𝑎𝑣𝑔
specimens switch from the brittle failure behaviour to ductile one. It was found that the
cyclic loading at confinement levels lower than the determined transition point has no
considerable effect on the post-peak instability of rocks, while for confinement levels of
80% and 100%, the weakening effect of the systematic cyclic loading history on rock
brittleness was significant. [see Chapter 8]
• According to the evolution of the tangent Young’s modulus (𝐸 ), cumulative
𝑡𝑎𝑛
irreversible axial strain (𝜔𝑖𝑟𝑟) and acoustic emission (AE) hits for hardening cyclic
𝑎
loading tests, it was observed that cyclic loading creates no macro-damage within the
specimens in the pre-peak regime, and the rock stiffness remains almost constant until
1000 loading and unloading cycles are completed. [see Chapter 8]
• For weakening cyclic loading tests (i.e., the tests that did not fail during the cycles and
showed negative peak strength variation), the gradual decrease and increase in 𝐸 and
𝑡𝑎𝑛
𝜔𝑖𝑟𝑟 were observed, respectively, with cycle loading. Moreover, compared to the
𝑎
hardening cyclic loading tests, the AE activities were more evident for specimens that
showed a higher amount of strength degradation. This is while for damage cyclic loading
tests (i.e., the tests that failed during cycles), the damage was accumulated with a higher
rate and extent in the specimens with an increase in confining pressure. [see Chapter 8]
• According to the variation of the axial strain at the failure point (𝜀 ) for the monotonic,
𝑎𝑓
hardening/weakening and damage cyclic loading tests, it was found that under
confinement levels below the transition point, the applied stress level (𝑞 /𝑞 ) has
𝑢𝑛 𝑚−𝑎𝑣𝑔
no significant effect on the cumulation of the plastic deformations in the pre-peak regime
and the values of 𝜀 are similar to those in monotonic loading conditions. However, for
𝑎𝑓
higher confinement levels, cyclic loading resulted in larger plastic deformations before
the failure point. [see Chapter 8]
• For the Gosford sandstone specimens that did not fail in cycles, it was found that the peak
strength varies between -13.18% and 7.82%. The strength hardening at lower confinement
levels, as observed for uniaxial systematic cyclic loading tests, can be related to the rock
compaction induced by cyclic loading. However, the increase in confining pressure
resulted in a decrease in strength hardening amount due to the accumulation of plastic
deformations in the specimens. [see Chapter 8]
• A CART-based model was proposed in this research project to estimate the peak strength
280 |
ADE | variation of Gosford sandstone as a function of the applied stress level (𝑞 /𝑞 ) and
𝑢𝑛 𝑚−𝑎𝑣𝑔
confinement level (𝜎 /𝑈𝐶𝑆 ). The coefficient of determination (𝑅2) for this practical
3 𝑎𝑣𝑔
model was 90% which proved the high prediction performance of this model. [see Chapter
8]
9.2. Recommendations
According to the methodologies used in this thesis and the corresponding obtained results, the
following recommendations are suggested for future studies to better address the rockburst-
related issues in deep underground mining operations:
• By considering the performance of the machine learning (ML) algorithms used in this
thesis (i.e., GA-ENN, C4.5, GEP, CART, and GEP-LR techniques) in dealing with high-
complex non-linear problems (e.g. rockburst hazard), establishing a more comprehensive
and precise rockburst database by including the intact rock properties, rock mass
parameters, geostress conditions, hydrogeological conditions and the geometry of the
excavations, holistic approaches can be developed to predict the rockburst occurrence and
its risk level accurately.
• Taking into account the well-known ML principle of “Garbage in, garbage out”, the
selection of the appropriate training datasets has a crucial effect on the reliability and
accuracy of the models. The ML-based rockburst models available in the literature have
been mostly developed based on the limited datasets (maximum 250 datasets, while almost
80% of them are considered for training the models). This is while the small amount of the
training samples cannot provide sufficient information for the ML algorithms, and finally,
the developed models may not be able to estimate the output parameter correctly by
feeding the new compiled datasets from the real projects. On the other hand, the available
rockburst databases in the literature are imbalanced, i.e., the number of data cases for each
rockburst risk level (i.e. “none”, “light”, “moderate” and “strong”) are not equal. This may
create biased models and decrease the applicability of the proposed models. Therefore, in
future studies, bigger and balanced databases should be provided to better analyse the
rockburst phenomenon. A promising technique to balance the database is over/under-
sampling.
• Many studies can be found in the literature regarding the rockburst potential evaluation in
the long term; however, no significant progress has been made in the short-term
assessment of this hazard using machine learning (ML) techniques. Microseismic signals
281 |
ADE | are significant precursors of rockburst occurrence. However, the genuine rock
microseismic signals usually interfere with the signals/noises induced by other sources,
such as mechanical excavations, haulage systems, drilling and blasting operations, etc. The
correct distinguishing of the genuine signals from the noise signals can provide some
critical features to estimate the rockburst occurrence. As proved in this research project,
the developed hybrid GEP-LR model is a powerful technique for multi-class classification
tasks and can be utilised in future studies to provide a practical model to discriminate
between different microseismic signals in burs-prone areas. By doing so, a proper
relationship can be established between the burst signals and rockburst occurrence, and
finally, the time of bursting can be predicted.
• The “Double-Criteria Damage-Controlled Test Method” developed in this research project
was recognised as an efficient methodology for capturing the post-peak behaviour of rocks
subjected to seismic events/cyclic loadings. In future studies, this technique can be
adjusted for the triaxial testing system to better analyse the failure mechanism of rocks
under different confining pressures.
• More in-depth numerical and experimental investigations should be undertaken
concerning the true post-peak behaviour of stable and unstable rock failures under
monotonic and cyclic loading conditions. In this regard, the influence of loading system
stiffness as well as the applied load control technique should be evaluated on the failure
behaviour of rocks. Although the lateral strain-controlled technique was identified as an
appropriate technique in capturing the complete stress-strain behaviour of rocks, the
capability of other load control techniques such as the linear combination of axial stress
(𝜎) and strain (𝜀) (i.e., 𝜀−𝛼.𝜎/𝐸 = 𝐶.𝑡, where 𝛼 is a constant less than 1.0, 𝐶 is the
loading rate which is usually set at 10-5/s and 𝐸 is Young’s modulus), which has been
reported as the potential load control technique in very few studies in the literature, needs
to be further investigated. It is also recommended to apply a large number of cycles (e.g.,
more than 1,000,000 cycles) in future rock fatigue studies to better replicate the seismic
events in real mining projects and evaluate its effect on fatigue threshold stress (FTS).
282 |
ADE | Abstract
The exploration and exploitation of hydrocarbon wells should not cause any environmental
hazards including contamination of groundwater (aquifers) and atmosphere. The cement placed in
the annular gaps between the casing strings and the formation acts as a key barrier to provide zonal
isolation and maintain the integrity of the wells.
The integrity of the cement sheath and the cement sheath interfaces is susceptible to be
compromised during well operational processes, including but not limited to, pressure integrity
tests (PIT), completion operations, stimulation treatments, and production processes. The cement
sheath may experience different types of mechanical damage as a result of being exposed to these
different wellbore operational procedures. Therefore, understanding of cement failure
mechanisms is of the utmost importance for better assessments of wellbore integrity.
This thesis demonstrates the results of the experimental-numerical studies and investigates the
integrity of the cement sheaths subjected to pressure and temperature variations.
The overall purpose of this study is to improve the modelling capabilities of cement sheath
integrity assessments by employing a more comprehensive constitutive model for the cement
sheath compared to the rest of the models previously used. The experimental studies on the
behaviour of the cement-based specimens under compression tests showed a strong non-linearity
in the obtained stress-strain curves which confirms the necessity of applying plasticity theories.
However, it is hard to explain the elastic stiffness degradation of the cement-based materials which
happens during experiments using the classical plasticity theories. Therefore, in this thesis, the
modified Concrete Damage Plasticity (CDP) model was employed, particularly formulated for
modelling geo-materials such as rocks, concrete, and cementitious materials. The Concrete
Damage Plasticity is a continuum model which combines plasticity and damage mechanisms,
considering two different tensile and compressive state of damage. The yield criterion in the CDP
model also represents the pressure-dependency of the geo-material behaviour under shearing at
different levels of confinement in addition to the incorporation of non-associated flow rule
(material dilatancy). These features show the superiority of the CDP model for employing in
cement sheath integrity assessments.
However, the paucity of cement class G mechanical parameters, e.g. lack of experimental data
under different confining pressure and tensile properties, was an impediment to the incorporation
of Concrete Damage Plasticity model.
I |
ADE | Therefore, the experimental aspect of this study intends to expand the cement class G inventory.
The experimental data and analyses added to inventory are as followed. The investigations of
curing temperature and pressure confinements effects on the strength and post-peak response of
the cement class G under compression and also obtaining cement tensile properties. The
experimental results show by increasing the curing temperatures, the compressive strength of the
material decreases significantly. This effect is attributed to the differences in the formation of
calcium silicate hydrate (CSH) gels due to an increase in the curing temperature. Additionally, by
increasing the confining pressure, the load-carrying capacity of specimens increases, and cement
shows more ductile behaviour. The results of three-point bending tests to obtain cement tensile
properties on prismatic samples showed that some modifications were required to be able to
measure cement fracture energy properly. Modifications were incorporated by employing the
crack-mouth clip gauge opening displacement to control the test loading rate, which led to less
brittle behaviour and allowed us to obtain the fracture energy. The results collected from the clip
gauge were validated by Digital Image Correlation (DIC) technique measurements. The
approximate shape of the yield surface for elastoplastic models was procured utilizing the
experimental data. The corresponding constitutive model parameters were computed by the curve
fitting process and were validated by numerical analyses. The incorporation of the obtained
parameters leads to the more accurate implementation of concrete damage plasticity model in
cement sheath integrity assessments.
In the numerical modelling aspect of this thesis, the integrity of cement sheaths was assessed
based on the local compression and tensile damage, and global damage indicators within the
cement sheaths considering different mechanical and thermal loading scenarios. The occurrence
of maximum compression and tensile damage on the narrowest side of the eccentric cement
sheaths confirms the importance of casing centralisation. The global damage indicator of
compressive (crushing) and tensile (cracking) states shows a higher possibility of cement sheath
failure while operating in anisotropic in-situ stress fields with soft rocks. The high magnitude of
tensile damage (cracking index) in some simulations confirms the importance of incorporating
tensile damage mechanisms into the constitutive modelling. The simulations result also showed
that cement sheaths subjected to controlled heating rates might experience less potential
compression damage comparing to cement sheath subjected to instant heating. The magnitude and
localisation of tensile damage were shown to be more dependent on the geometry of the wellbore
rather than the heating rates. In cooling scenarios, the effects of wellbore contractions due to
II |
ADE | Statement of originality
I certify that this work contains no material which has been accepted for the award of any other
degree or diploma in my name, in any university or other tertiary institution and, to the best of my
knowledge and belief, contains no material previously published or written by another person,
except where due reference has been made in the text. In addition, I certify that no part of this
work will, in the future, be used in a submission in my name, for any other degree or diploma in
any university or other tertiary institution without the prior approval of the University of Adelaide
and where applicable, any partner institution responsible for the joint-award of this degree.
I acknowledge that copyright of published works contained within this thesis resides with the
copyright holder(s) of those works.
I also give permission for the digital version of my thesis to be made available on the web, via
the University’s digital research repository, the Library Search and also through web search
engines, unless permission has been granted by the University to restrict access for a period of
time.
I acknowledge the support I have received for my research through the Completion
Scholarship.
Signature: Date: 11-12-2018
V |
ADE | 1. Introduction
There are more than four million onshore hydrocarbon wells drilled worldwide [9] with nearly
10000 in Australia alone [9] (from data retrieved from Geoscience Australia). A wellbore provides
access to natural sources such as oil and gas. The wellbores are encased in different layers of steel
casings and cement sheaths within the annuli. After drilling, the steel casing is run into a borehole,
and be placed and protected with the help of Portland cement. The cement is placed by cement
slurry circulation downward through the central wellbore and up the annular space between the
casing and the rock. Layers of the casing with decreasing diameters are placed at the centre of the
hole in each step [10-12]. The number of casing strings is dependent on the formation properties
for each wellbore. In general, a well can have between two to four casing strings including the
conductor, surface, intermediate, and production casing as shown in Figure 1.1. These casing
strings run to different depths, and one or two of them may not be required based on the drilling
conditions. These strings might be run as liners or in combination with liners [1].
Conductor casing is the first layer of casings with the largest diameter running from the surface
until the depth of 12-150 meters in onshore wells and up to 300 meters in offshore wells [1].
Conductor casing prevents any unconsolidated surface sediments to enter the wellbore [1]. Surface
casing is placed after conductor casing is installed and cemented. The length of surface casing
varies according to each well design and can be up to 1500 meters. One of the roles of surface
Cement Sheath
Conductor Casing
Surface Casing
Intermediate Casing
Production Casing
Figure 1.1: Wellbore Architecture after [1]
1 |
ADE | casing is isolating the freshwater-bearing formations [1]. The intermediate casing or protective
casing is placed between the surface casing and production casing to protect any unusual high-
pressure rock from initiating wellbore instability. An additional layer of intermediate casing might
be required corresponding to the different formation characteristic, i.e. abnormal formation pore
pressure. Intermediate casing varies in length from 2000-4500 meters [1]. The Production casing
is run as the final casing, and it starts from the surface to the reservoir, and it protects the
prospective productive zone from other subsurface formations [1, 13].
The exploration and exploitation of hydrocarbon wells should be in line with the protection of
the environment to prevent groundwater (aquifers) contamination [14, 15] and migration of
fugitive emissions [16] into the atmosphere [9]. Groundwater sources are protected from the
contents of well operational processes, i.e. drilling, hydraulic fracturing, production operations,
etc. by layers of steel casing, and cement sheaths which act as multiple barriers to separate the
formation fluids from the outside environment [17]. Although wellbores are sealed and prevent
any communication between formation fluid and geologic strata (which may contain
groundwater), the integrity of wellbores might still be compromised [9, 18]. At this point,
wellbores may turn into the high-permeability conduits for the formation fluids [19] which impose
a potential risk to the environment by polluting the groundwater and atmosphere. To maintain the
integrity of the wellbores, a wellbore barrier system should be designed in a way to endure the
mechanical and thermal operational procedures imposed by production and recovery phases
during a wellbore lifetime.
However, wellbore barrier failure might occur due to the failure of the individual or multiple
barriers even if there are no indications of detectable leakage into the wellbore surroundings [18].
If a barrier fails, an assessment has to be done to evaluate the imposed risk of fluid leakage and
repairing procedures should be planned. A barrier failure might happen during different stages of
a wellbore lifetime, i.e. pre-production phases / and production phases [20].
1.1. Barrier Failure during Pre-Production Phase
Some of the well operational procedures may lead to a barrier failure in the pre-production
phase, i.e. pressure integrity tests (leak-off tests) [21-24], extended leak-off tests [25]. Pressure
integrity tests (PIT) are performed after the cementation of each casing, and impose pressure upon
set cement [26]. Drilling practices may also damage the unstable formations (caving) due to the
imposed vibrations and pressures which may lead to formation failure. In addition, some
formations are naturally weak and not stable enough or may have some faults and cracks. These
2 |
ADE | faults can threaten the integrity of the wellbores even before the commencement of production
procedures [20, 27].
The casing centralization should be executed properly. Otherwise, the cement would not be able
to move the mud from the annulus completely during cementing procedures and leads to the
formation of eccentric cement sheath and non-uniform cement sheath thickness or possibly not
fully covers the created gap [20]. This deviation of the casing from the centre can cause unbalance
concentration of stress on the one side of the wellbore which results in additional shear stress to
the cement sheath [28].
The existence of mud cake and grease deteriorates the bond strength between the cement with
the casing and the formation during cement pumping procedure. Additionally, contamination of
cement by mud or formation fluid may weaken the cement mechanical properties as well, which
may lead to compromising the wellbore integrity. [20, 29]. Muds have a thixotropic behaviour and
tend to build a gel-structure under low shear circumstances. The gelled pockets should be broken
up and cleaned to achieve a stronger cement bonding. Another reason could be related to the
improper composition (cement slurry formulation) of the cement slurry, in terms of its
compatibility with the formation which results in weak bonding properties [12, 20].
Cement shrinkage leads to a volumetric reduction and can consequently cause de-bonding
between cement and casing or formation. This can also result in tensile cracks and increased
permeability which provides pathways for undesired fluid and gas migration. [12, 20, 30].
Due to high-pressure conditions (high gradient of pressure between the well and the formation),
the fluid in the cement slurry could be filtrated. This lack of water during the hydration process
will decrease the cement strength [20].
1.2. Barrier Failure during Production Phase
During production phases, the mechanical and thermal stress state of a wellbore is subjected to
different pressure and temperature variations due to different reasons [31] including the alteration
in induced pressure and temperature originating from casing expansion / contraction [32],
hydraulic stimulation [33], loading from formation stresses such as tectonic stress, subsidence and
formation creep [12], change of pore pressure or temperature [34], normal well production [12],
injection of hot steam or cold water [35, 36], etc. These operational procedures have significant
effects on the integrity and the failure mechanism of cement sheaths.
Wellbore integrity failure might occur when all the wellbore barriers fail, and leakage pathways
are created. Thus, leakage is detected in the soils, strata, and or atmosphere [18]. Potential leakage
pathways might already exist or be created in different regions within these multiple barriers
3 |
ADE | system during the lifetime of a wellbore [10]. Leakage paths are divided into two categories,
primary and secondary. Primary category is more related to the time of primary cementing and
secondary are associated with the events and conditions after cementing is complete [6].
Figure 1.2 illustrates the possible locations of primary and secondary leakage pathways along
a wellbore. Primary leakage pathways can be created due to casing burst or collapse (Figure 1.2b)
[6, 10], unsatisfactory annular cementing job when the cement does not fill the annulus entirely
(Figure 1.2f), poor bonding due to the existence of mud cake (Figure 1.2g), and development of
channels in the cement (figure 1.2d) [6].
Casing
a)
Cement Sheath
Formation
f)
b)
g)
c) d)
e)
Figure 1.2 : Potential Leakage Pathways along a Wellbore after [6, 7]
The secondary category included the leakage pathways created along micro-annuli at the
cement sheath interfaces with the casing and the formation respectively [7, 10] as shown in Figure
1.2a and Figure 1.2e, and degraded or cement fractures (Figure 1.2c) [6, 10].
These pathways might be created due to many reasons including but not limited to deterioration
of cement bond strength which leads to the creation of micro-annulus at cement interfaces with
the casing and the formation, poor removal of the mudcake formed during drilling, mechanical
failure of the casing, cement shrinkage, and cement mechanical failure [34].
4 |
ADE | The cement sheath is subjected to variations of mechanical and thermal cycles due to different
wellbore operational processes, i.e. drilling, hydraulic fracturing, production operations, etc.
during the lifetime of a wellbore. Hence, the integrity of the cement sheath and the cement sheath
bonding integrity [37] affect the long-term integrity of the wellbores significantly [6].
It is worth noting that the cement used in the oil and gas industry has very low permeability,
usually less than a 0.2 mD [38] which indicates that hydraulic isolation is accomplished
straightforwardly, and any possible leakage can only occur through mechanical failures of the
cement sheath [11]. Therefore, the integrity of the cement sheath may be compromised mostly
because of the creation of cracks and micro-annulus within the cement sheath [39].
1.3. Cement Sheath Serving as the Key Barrier
Well-cementing (cementation) is an influential stage of a wellbore completion since the cement
sheath is responsible for providing complete zonal isolation [40]. The cement sheath should meet
both short-term and long-term required characteristics to overcome all pressure and temperature
variations imposed to a well during well lifetime and also after it is decommissioned/abandoned
[41]. Accordingly, it is of utmost importance to comprehend the cement mechanical failure
mechanisms. The cement sheath may experience different types of mechanical damage as a result
of exposing to different wellbore operational procedures [8].
It is worth noting that cement class G is mostly utilised in the oil and gas industry. Class G is
manufactured by implementing the improved technology in slurry acceleration and retardation
with respect to their chemical reactions. Manufacturers are not allowed to add special chemicals,
including glycols or acetates, to the clinkers [42]. These chemicals enhance the grinding efficiency
but have been shown to intervene with the effect of various cement additives. Classes G and H are
the most commonly employed in well cements nowadays. Class G is mostly utilised as a basic
ingredient for cementing from the surface to 8,000 ft (2,440 m) depth as manufactured or can be
employed along with accelerators and retarders to cover a wide range of well depths and
temperatures [42].
Figure 1.3 schematically demonstrates the different types of cracks may occur within the
cement sheaths. Radial cracks (Figure 1.3a) might be created due to the difference in pressure
between the inner wall of the cement sheath with the outer wall which leads to the cement sheath
expansion/contraction [8]. The cement sheath may experience a large deviatoric state of stress
which leads to shear damage (Figure 1.3b) [8]. Disking cracks might be created due to axial sliding
/ disking of the cement sheath (Figure 1.3c) [8]. The cement sheath interfaces debonding may
occur due to the uneven expansion/contraction of the cement sheath in comparison with the
5 |
ADE | c) Disking Cracks
a) Radial Cracking b) Shear
d) Interface debonding
Figure 1.3: Different Types of Cracks within the Cement Sheath after [6, 8]
displacement of the surrounding wellbore components which leads to the creation of micro-
annulus within the wellbores (Figure 1.3d) [8]. Consequently, understanding of cement failure
mechanisms under different operating conditions is of the utmost importance for the better
evaluation of wellbore integrity.
Mechanical failure of the cement sheath within a wellbore is affected and governed by many
factors including material mechanical properties (cement compressive strength [4-6], Young’s
modulus [43-45], tensile strength [31, 46], and bond strength [37, 47]), loading conditions (in-situ
stresses [44, 46, 48]), cement history (cement shrinkage) [8], and also wellbore architecture
(cement sheath thickness, formation properties, cement sheath eccentricity, and wellbore deviation
[8, 37]).
A comprehensive model is required to consider the contribution of each aforementioned factor
in predicting the initiation and propagation of the cement mechanical failure. So far, different
analytical and numerical modelling approaches were carried out to achieve a better assessment of
cement sheath integrity in wells. Numerical modelling, including Finite Element Method (FEM)
in particular, has been considerably improved regarding their accuracy and ability to incorporate
different constitutive models, complex types of geometry and boundary conditions, and in-situ
stress conditions [47]. The incorporation of appropriate material constitutive law and subsequently
the evolution of corresponding model parameters are fundamental stages in order to develop a
numerical model.
6 |
ADE | To this point, the linear elastic was employed in a few cement integrity analyses, i.e. [28, 37,
49]. However, the obtained stress-strain curves from the isotropic drained compression tests on
the cementitious specimens by [36] are non-linear. Therefore, the employment of linear elastic
theory in cement integrity simulations perturbs the accuracy and reliability of the results.
Additionally, the existence of the permanent strains upon unloading [36] confirms the
incompatibility of linear elastic theory in cement integrity evaluations again as the elastic theory
does not incorporate the time-dependency and materials hysteresis law [50].
The non-linear approaches including those employing Von-Mises [51], Mohr-Coulomb / with
smeared cracking [31, 44], Drucker-Prager [52], Ottosen model [5, 53], and modified Cam-Clay
[36] were incorporated in the cement sheath integrity assessments to alleviate the shortcomings of
the linear elastic models. These approaches along with their merits and limitations are reviewed
in the literature review Section 3.1.2. completely.
Notwithstanding all the progress has been made in the cement integrity simulations in numerical
fields, some aspects of the modelling still require attention including the incorporation of a
comprehensive constitutive which reflects both compression and tensile damage mechanisms in
addition to the pressure-dependency of the behaviour subjected to confining pressures.
Furthermore, the incompleteness of the cement (cement class G utilised in oil and gas industry)
mechanical parameters inventory is another impediment to the numerical modelling, for instance,
the function of triaxial tests experiments to approximate the shape of the shape of yield / and
failure surfaces are neglected. The measurement procedures of cement tensile strength properties,
fracture energy, in particular, are not consistent. However, these properties are required to simulate
the tensile behaviour mechanism. Moreover, the effect of curing temperature on the cement long-
term mechanical properties was also missing from the literature to the best of author’s knowledge.
The experimental studies available in the literature are explored and reviewed in section 3.2. along
with their merits and limitations.
Consequently, in this research, the emphases were placed on filling the gaps in the cement
mechanical properties inventory in addition to the incorporation of an appropriate constitutive
model (Concrete Damage Plasticity model) specifically formulated for the modelling of geo-
materials developed and modified by [54, 55].
The main advantage of Concrete Damage Plasticity (CDP) model is coupling plasticity with
damage mechanism which evidently describes the elastic stiffness degradation of materials during
the experiments due to the creation of microcracking. The creation of microcracks which is also
characterized by softening behaviour of the materials is difficult to explain using classical
7 |
ADE | plasticity models [55]. The modified version of CDP by [55] benefits from considering the
difference in tensile and compressive responses of geo-materials since geo-materials experience
different states of damage while subjected to different loading conditions. This model also
considers the materials pressure-dependency behaviour under shearing at different levels of
confinement. The non-associated flow rule which represents the dilatancy of the geo-materials
also embedded into the model. These features make this model a very suitable model to be applied
to a range of geo-materials including rocks, and cement-based materials [55].
To be able to incorporate the CDP model into the simulations the corresponding constitutive
model parameters were determined by experimental investigations along with parameters
calibrations to ensure their reliability for cement sheath integrity assessment.
In the experimental aspects of this thesis, laboratory experiments including confined and
unconfined compression tests, and three-point bending tests considering different curing
conditions were performed on specimens manufactured from class G well cement. The
approximate shape of the yield surface for elastoplastic models was obtained using the
aforementioned experimental data, and the corresponding parameter intended for Concrete
Damage Plasticity model was computed by calibration process and were also validated by
numerical analyses. The incorporation of the obtained parameters leads to the more accurate
implementation of Concrete Damage Plasticity model into the cement sheath integrity
assessments.
In the numerical aspects, three-dimensional (3-D) finite element frameworks are developed
employing the constitutive model for cement sheath and a surface-based cohesive behaviour for
the interfaces in the cement sheath integrity investigations. The obtained parameters from the
experimental aspect of this thesis implemented into the Concrete Damage Plasticity model for the
cement sheath subjected to variations of mechanical and thermal loads. The effects of anisotropy
of in-situ stresses, different stiffnesses of surrounding rocks, and different degrees of cement
sheath eccentricity within the wellbores on the integrity of the cement and interfaces are also
investigated.
Moreover, the outcomes of numerical models are mesh dependent which might be a source of
uncertainty within the integrity simulations. To lessen the drawbacks of mesh dependency in
numerical analyses the concepts of crack band methodology by Bažant and Oh [56] was applied
through incorporation of the characteristic length [57] in utilised software (ABAQUS) which is
related to the element size and formulating the softening part of the constitutive law by embedding
the stress-displacement instead of stress-strain relationship. Incorporating the stress-displacement
8 |
ADE | 2. Thesis Overview
This thesis is organised into nine chapters where the main contributions are presented in
Chapter 5 to Chapter 7. Each of these chapters is presented in the form of a technical paper. The
first of these has been published in the Journal of Petroleum Science and Engineering, the second
has been published in Australian Journal of Civil Engineering. The third paper is well-prepared
and will be submitted for peer review shortly.
In Chapter 3, a literature review was explored on cement sheath integrity modelling and
experimental laboratory studies. The advantages and limitations of different approaches in the
literature were assessed which leads to the identification of the research gaps.
Chapter 4 explains the overall objective of this research along with the three specific objectives
and the linkage between the research objectives and papers.
In Chapter 5, an experimental-numerical study is represented to investigate the effect of
enhancing pressure on the cement sheath integrity. Concrete Damage Plasticity constitutive model
specifically formulated for the modelling of geo-materials was applied to the investigations of
cement sheath integrity, incorporating both compression and tensile damage mechanisms.
Laboratory experiments were carried out to obtain strength properties of cement class G followed
by calibration of the model parameters based on the obtained experimental results. A three-
dimensional finite element framework employing the constitutive model for cement sheath and a
surface-based cohesive behaviour for the interfaces was developed for integrity investigations.
The effects of different orientations and the anisotropy of in-situ stresses, different stiffness’s of
surrounding rocks, and the eccentricity of the casing within the wellbore on the integrity of the
cement and interfaces were investigated.
Chapter 6 describes the laboratory experiments that were carried out to investigate the effect
of curing conditions on the cement class G mechanical properties, including confined and
unconfined compression tests and three-point bending tests on specimens cured at different
conditions. The interpretation of the results and experimental parameters calibration and validation
were performed to ensure their suitability to predict the behaviour of cement class G.
In Chapter 7, a numerical approach was undertaken to investigate the integrity of eccentric
cement sheaths after being subjected to mechanical and thermal wellbore operational procedures
in relation to the creation of cracks within the cement sheath. The importance of incorporating the
appropriate constitutive model (Concrete Damage Plasticity model) for modelling geo-materials
such as well cement was highlighted. Three-dimensional finite element frameworks employing
the constitutive model for the cement sheath and a surface-based cohesive behaviour accompanied
11 |
ADE | 3. Literature Review
The importance of applying a comprehensive model to simulate the cement sheath behaviour
under downhole conditions has been highlighted throughout the introduction chapter. In the
following sections, the cement sheath numerical modelling and the cement experimental studies
are reviewed.
3.1. Cement sheath Modelling
Cement sheath integrity models investigated could be categorised into analytical and numerical
models. Analytical methods are generally performed by applying simplified assumptions to
facilitate finding solutions. The accuracy of analytical models and subsequently their solutions are
limited to the correctness and the suitability of their initial assumptions and simplifications [47].
While numerical modelling can be very advantageous considering its ability to incorporate
material non-linearity, different types of geometry and boundary conditions, and in-situ stress
conditions [9], the accuracy of these numerical models is reliant on the validation and verification
of obtained experimental data utilised as inputs for constitutive models [47].
Different wellbore operations including sudden dynamic loading [59], perforation of the casing
[11], CO injections [60], hydraulic fracturing [61], acidization and finally production of the
2
reservoir, variations of production rate [37], pressure integrity tests (leak-off tests) [24] affect the
stress distribution within the cement sheath and the cement sheath bond with the casing and the
formation. During pressure testing, fracturing and acidizing and normal production, the wellbore
will be pressurised which may lead to different failure mechanisms (compression/shear) to the
cement sheath. The tensile failure may also happen as the results of high contact shear stress at
the interfaces of the cement sheath with the casing and the rock formation due to pressuring or de-
pressuring the wellbores. CO2 injections may cause thermally induced expansion and contraction
within the wellbores, possibly resulting in the formation of leakage paths. The thermal loading
and unloading generate thermal stresses inside the wellbore components [62]. Consequently, the
well barrier materials may fail as a result of the thermal cycling operations.
Cement Sheath: Analytical Modelling
Thiercelin, Dargaud, Baret and Rodriquez [63] developed a plane strain analytical approach to
measure the induced damage and determining of controlling key parameters assuming the linear-
elastic properties for cement, axisymmetric geometry, and fully bonded or unbound situations for
13 |
ADE | the interfaces. Their results showed the mechanical response of the set cement is dependent on the
mechanical properties of the cement and the rock, and wellbore geometry.
Honglin, Zhang, Shi and Xiong [64] have proposed a 2-Dimensional (2-D) analytical model
using Mohr-Coulomb failure criterion to investigate the effect of well head casing pressure
(WHCP) on the cement sheath integrity in high pressure and temperature (HPHT) wellbores. They
suggested a safety factor diagram considering different ranges of temperature and WHCP at the
casing interface. Their safety factor diagram showed in the circumstances with high WHCP, the
influence of temperature change on the cement sheath failure was diminished which also results
in low safety factors. They stated that in cases with WHCP below 40MPa, the effect of temperature
can generally be neglected.
Shi, Li, Guo, Guan and Li [65] estimated the initial radial and tangential stresses at cementing
interfaces with the assumption of axisymmetric geometry, isotropic horizontal in-situ stresses and
elastic properties for the cement sheath and interfaces.
Zhang, Yan, Yang and Zhao [52] proposed an analytical plain-strain approach to assess the
integrity of a wellbore under HPHT conditions by coupling solid-temperature approach. The
Mises criterion, Drucker-Prager, and Joint Roughness Coefficient-Joint Compressive Strength
(JRC-JCS) were exploited to model the casing, cement sheath and cement interfaces respectively.
In their parametric study, they showed that the cement mechanical properties affect the failure
coefficient of the casing-cement sheath-formation system significantly. They tried a wide range
of 3 GPa to 90 GPa for the cement Young's modulus and 0.1to 0.4 for the cement Poisson ratio.
They demonstrated that incorporating cement with low Young's modulus and high Poisson ratio
resulted in lower failure coefficient, therefore, it is more favourable to the wellbore integrity.
The assumptions and simplifications made in analytical models such as the aforementioned
study may lead to unrealistic results. For instance, failure modes in all directions would not be
captured in two-dimensional (2-D) plane strain models. Furthermore, the axisymmetric geometry
and the assumed isotropic in-situ stresses do not correctly reflect the real conditions [47].
Cement Sheath: Numerical Modelling
Numerical modelling has been significantly improved compared to analytical modelling
regarding complexity and ability to model wellbore integrity assessment with a high degree of
accuracy. The incorporation of appropriate material constitutive law and consequently the
evolution of corresponding model parameters are the fundamental stage in developing a numerical
model.
14 |
ADE | To this point, the elastic linear principle was utilised in a few cement integrity studies reviewed
as follows. Nabipour, Joodi and Sarmadivaleh [28] simulated downhole stresses using Finite
Element Method (FEM) along with sensitivity analyses on casing internal pressure, anisotropic
horizontal in-situ stresses, and casing eccentricity. They have used a plain strain model with
thermo-elastic material properties, and the interfaces are assumed to be fully bonded. According
to this study, the failure of cement and formation bond and the initiation of radial cracks from the
inner surface of the cement sheath are the most possible scenarios for losing the cement sheath
integrity.
Wang and Taleghani [37] performed a three-dimensional (3-D) poroelastic simulations with a
particular focus on the interface modelling to assess the integrity of the interfaces. They also
explained the superiority of developing 3-D models in terms of capturing the spatial fracture
patterns which may not be completely explained by common two-dimensional axisymmetric
models. Since in these 2-D models the failure paths are constrained to the direction parallel to the
borehole axis.
Guo, Bu and Yan [49] presented a numerical study to investigate the effect of the heating period,
cement thermal expansion, and overburden pressure on the cement integrity under steam
stimulation conditions. All materials presumed to be linear elastic. They recommended a moderate
heating rate and moderate cement thermal expansion coefficient is beneficial to the cement sheath
integrity.
Li, Liu, Wang, Yuan and Qi [48] developed a coupled framework to investigate the effect of
non-uniform in-situ stress filed, temperature, and pressure effects on wellbore integrity. The stress
states evaluated assuming the linear elastic behaviour for all the materials. According to this study,
the anisotropy of in situ stresses resulted in the creation of shear stresses and non-uniform stress
distribution within the cement sheath. By increasing the casing temperature, the tensile stresses
develop and lead to the creation of fractures in the inner surface of the cement sheath.
De Andrade and Sangesland [66] conducted a numerical study with a special focus on thermal-
related load cases. They built a 2-D model and assumed a linear elastic behaviour for all the
materials, bonded contact between wellbore components and isotropic in-situ stresses. A
utilisation factor based on Mogi-Coulomb criterion was defined to check the state of the stress and
estimate cement sheath failure. The utilisation of Mogi-Coulomb criterion instead of Mohr-
Coulomb was explained by considering the obtained experimental data by Al-Ajmi [67] which
states Mogi-Coulomb criterion represents the state of shear failure in different types of rocks better
than Mohr-Coulomb criterion. According to their results, the likelihood of cement sheath damage
15 |
ADE | and bonding failure is higher in cooling scenarios compared to the heating scenarios. The effect
of casing centralisation and controlled heating/cooling rates seemed to be trivial.
The employment of linear elastic theory to simulate the cement sheath behaviour can affect the
accuracy and reliability of the results due to the oversimplifications made in finding solutions. The
complex response of the cement to different mechanical and thermal loading scenarios cannot be
simulated by elastic theory. The obtained stress-strain curves from the isotropic drained
compression tests on the cementitious specimens by Bois, Garnier, Rodot, Sain-Marc and Aimard
[36] clearly indicate non-linear behaviour. In addition, the existence of the permanent strains upon
unloading [36] confirms the incompatibility of linear elastic theory and the necessity of employing
plasticity theory in cement integrity evaluations again. Considering that, the elastic theory doesn’t
incorporate the time-dependency and materials hysteresis law [50].
The non-linear approaches including Von-Mises [51], Drucker-Prager [52], Ottosen model [5,
53], modified Cam-Clay [36], and Mohr-Coulomb / with smeared cracking [31, 44] were
incorporated in the cement sheath integrity assessments to lessen the drawbacks of the applied
linear models.
Fleckenstein, Eustes and Miller [51] employed the von-Mises criteria and showed that the
magnitude of tangential stresses would be significantly reduced if the cement sheath acts as a
ductile material with lower Young’s modulus and higher Poisson’s ratio. The lack of pressure
dependency of the von Mises criteria is however problematic in modelling cementitious materials.
Zhang, Yan, Yang and Zhao [52] utilised Drucker-Prager failure criterion in a 2-D model to
verify their proposed analytical model. Pattillo and Kristiansen [68] also employed Drucker-
Prager criterion on their 2-D model to investigate the integrity of Valhal horizontal wellbores. In
both studies, the sources of constitutive model parameters are not detailed. The studies carried out
on the performance of Drucker-Prager model shows this model does not provide accurate
predictions while one or more principle stresses are tensile stress. Additionally, considering the
same effect for 𝜎 and 𝜎 leads to overestimation of rocks’ strength and it is not verified by
2 3
laboratory experimental data [69, 70].
Asamoto, Le Guen, Poupard and Capra [5], Guen, Asamoto, Houdu and Poupard [53]
developed a 2-D model using the Ottosen model [71] as a smeared crack model to investigate the
softening post-peak behaviour of the cement sheath and the estimation of the crack width in a
wellbore subjected to thermal and mechanical loads. In both studies, the details of the constitutive
model performance and the relevance of the constitutive parameters to the experimental data are
not described.
16 |
ADE | The modified Cam-Clay model has been suggested as a method to incorporate cement micro
cracking mechanisms by Bois, Garnier, Rodot, Sain-Marc and Aimard [36] owing to the
nonlinearity of stress-strain curve achieved from the isotropic drained compression tests [72] and
heterogeneous nature of cement at the microscale. Although important aspects of materials
behaviour (material strength, compression or dilatancy, and critical state of elements under high
distortion) are considered in this model, the tensile post-peak material is not incorporated into this
framework.
Mohr-Coulomb criterion alone or combined with the smeared cracking model has been used in
some studies which are reviewed as follows. One of the models was proposed by Bosma, Ravi,
van Driel and Schreppers [44]. They developed a 2-D model considering symmetry geometry for
the wellbore. Mohr-Coulomb plasticity combined with smeared cracking description was used to
model the cement sheath under compression/shear and tension. The cement sheath interfaces were
modelled using interface elements applying a coulomb friction criterion. According to this study,
considering only the cement failure envelope in compression as a quality indicator is not
acceptable in wellbore integrity modelling. The cement Young’s modulus, Poisson’s ratio, tensile
strength, shear strength, and bonding properties are to be incorporated into the wellbore integrity
modelling.
Ravi, Bosma and Gastebled [31] developed a 2-D model to investigate the wellbores integrity
subjected to operational procedures. To model the stress state within the cement sheath, the
Hookean model was incorporated for undamaged state and combined Mohr-Coulomb plasticity
with smeared cracking after exceeding the compressive shear and tensile strength state. According
to their findings, the integrity of the cement sheath is highly dependent on the cement and
mechanical rock properties, and well-operating parameters. Cement sheath with less stiffness
shows more resilient and helps to reduce the risk of cement sheath failure. Petty, Gastineau, Bour
and Ravi [73] also used Mohr-Coulomb plasticity combined with smeared cracking in their 2-D
model to determine the advantageous cement system with respect to the integrity of the cement
sheath within a geothermal well. They showed that foamed cement performs better than the
conventional cement while being exposed to pressure-temperature stresses and the effect of
shrinkage is also minimised by using foamed cement.
Mohr-Coulomb criterion was also used by Feng, Podnos and Gray [50], Nygaard, Salehi,
Weideman and Lavoie [74], and Zhu, Deng, Zhao, Zhao, Liu and Wang [75] to predict the plastic
behaviour of the cement sheaths subjected to mechanical and thermal loads.
17 |
ADE | The combination of Mohr-Coulomb with smeared cracking is one of a few suitable approaches
for modelling the real conditions in the cement integrity numerical simulations. However, despite
the broad application of Mohr-Coulomb criteria, it has its own limitations. The model assumes a
linear relationship between √J and I in the meridian plane, while this relationship has been
2 1
experimentally shown to be curved [36, 72, 76, 77], for cementitious materials, particularly at low
confinement. The major principal stress 𝜎 and intermediate principal stress 𝜎 are defined
1 2
independently in Mohr-Coulomb model which results in an underestimation of the yield strength
of the material and, it is not in a good agreement with experiments in which the effect of 𝜎 is
2
being considered. The shape of the yield surface in the deviatoric plane is an asymmetrical
hexagon, whereby the sharp corners can hinder convergence in numerical simulations [70, 78].
Moreover, quasi-brittle materials experience a huge volume change due to a large amount of
inelastic strains (dilatancy) which has been overlooked so far by using associated flow rules in the
aforementioned modelling approaches of the cement. The associative plastic flow rules tend to
lead to poor results in dilatancy evolution [55].
The application of plasticity theory in compression (Mohr-Coulomb) combined with the
fracture mechanics models such as smeared cracking presents some drawbacks as well. Given
that, smeared crack models in finite element analysis can often be problematic in terms of “mesh
alignment sensitivity” or “mesh orientation bias” which indicates that the orientation of smeared
crack depends on the discretization orientation [79]. It is worth adding that the mesh regularization
approach proposed by [56] (crack band theory) in the smeared cracking model has been successful
for predicting mode I fractures while the extension of this approach to mixed-mode failure and
three-dimensional stress state is hard [79].
Considering the limitations, it would be more practical to employ more suitable models with
respect to their accuracy (enrichment) and reliability (capability to reproduce the experimental
data) along with their efficiency (mesh orientation and mesh size objectivity) [79].
The observed non-linearity in cement behaviour [36, 72] results from two different
microstructural changes which happen in the materials while subjected to different loading
conditions. One is plastic flow causes the permanent deformation and the second is the
development of microcracks which leads to elastic stiffness degradation [80]. Therefore, it a
necessary to apply a model which combines plasticity and damage mechanics. For this reason, the
formation of microcracks which is also characterized by softening behaviour of the materials is
difficult to explain using classical plasticity models [55]. The damage mechanism is described by
two physical aspects corresponding to the two modes of cracking ( hardening and softening) [80].
18 |
ADE | Therefore, in this study, Concrete Damage Plasticity (CDP) model developed by [54] and then
modified by [55] has been employed. The Concrete Damage Plasticity model combines plasticity
and damage mechanics and uses the concept of fracture-based damage. In the modified revision,
two damage variables one for compressive damage and one for tensile damage incorporated to
consider different states of damage. This feature is capable of describing the induced anisotropy
of microcracking which also facilitates the numerical implementation procedures [80]. The
pressure-sensitive yield criterion accompanied by employing the dilatancy (non-associated flow
rule), makes this model more suitable than the others that have been employed in the assessment
of cement sheath integrity.
Cement Sheath Interfaces Modelling
The cement sheath interfaces with the casing and the formation are recognized as the weakest
link and the most potential area for defects and debonding issues which leads to losing the cement
sheath integrity [37, 81].
The delamination mechanism is one of the most uncertain aspects of wellbore integrity
simulations due to its complex nature [82]. So far, different modelling approaches have been
undertaken to simulate the behaviour of the cement sheath interfaces. Bosma, Ravi, van Driel and
Schreppers [44] modelled the behaviour of the interface by interface elements using the Coulomb
friction failure criterion, and the elastic stiffness of the contact elements was chosen considerably
higher than that of the surrounding material. Guen, Asamoto, Houdu and Poupard [53] employed
Mohr-coulomb failure criterion for the interface modelling. The details of obtaining the process
of the corresponding parameters for the interfaces are not elaborated upon. The use of Mohr-
Coulomb criterion may not be substantially appropriate due to the complicated behaviour of the
interfaces. The delamination may occur at the mixed-mode conditions and not only within pure
compression/shear condition.
Zhang, Yan, Yang and Zhao [52] incorporated the non-linear criterion knows as Joint
Roughness Coefficient- Joint Compressive Strength (JRC-JCS) which was originally developed
by Barton and Choubey [83] for joint rock analysis. The corresponding parameters were taken
from the literature from the rock analysis which may not be accurate to be used at the cement
sheath interfaces modelling.
The most recent and successful cement sheath interface modelling was carried out by Wang
and Taleghani [37]. They modelled the interfaces by incorporating the cohesive theory to simulate
the initiation and propagation of debonding at the interfaces. They performed an inverse analysis
on the experimental results performed by Carter and Evans [84], Ladva, Craster, Jones, Goldsmith
19 |
ADE | and Scott [85], Evans and Carter [86] to determine the corresponding parameters of the cohesive
criterion.
Evans and Carter [86] designed a push-out test setup to measure the cement shear bond and
hydraulic bond to the casing and the formation. Carter and Evans [84] continued their
experimental work and identified more influential factors on the cement bonding properties to the
casing. They designed cylindrical chambers in which shear bond is determined by applying force
to instigate the movements of the casing surrounded by cement. The shear bond measured as
dividing the force to the contact surface area. The cement hydraulic bond is determined as the
cement bond to the casing and the formation which prevent the fluid migration. Hydraulic bond is
determined by applying pressure to the cement interfaces until leakage happens.
As mentioned, the approach proposed by Wang and Taleghani [37] to model the cement sheath
interfaces behaviour with cohesive elements has been very successful in numerical simulations.
Therefore, their approach has been applied in this thesis along with minor alterations. In this
research. the interfaces are modelled using “surface-based cohesive behaviour” instead of
cohesive elements. The surface-based cohesive behaviour defines as a surface interaction property
with traction transferring capacity between master and slave surfaces. The cohesive constraint is
enforced at each slave node for cohesive surfaces. Therefore, for cohesive surfaces, refining the
slave surface in comparison with the master surface will result in the improved constraint
satisfaction and more accurate results than using cohesive elements [87]. In addition to providing
mesh generation flexibility at each side of the interfaces. Moreover, the employment of surface-
based cohesive behaviour instead of cohesive elements complies with the incorporation of the
temperature transmitting capabilities at the interfaces to overcome the limitation of the
nonexistence of temperature degree of freedom in cohesive elements.
3.2. Laboratory Experiments on Cement Properties
In 1992, Goodwin and Crook [32] performed laboratory investigations to simulate conditions
at which the cement sheath failure occurs. They built a prototype consisting of an inner casing,
outer casing, and the annulus filled with cement. They observed sudden exposure to excessive
internal pressure and temperature result in radial and circumferential casing expansion. The
diametrical and circumferential forces create radial and shear forces within the cement sheath and
at the interface of cement with the casing.
Afterwards, the researchers attempted to incorporate the predictive models to simulate the
failure scenarios. However, the incompleteness of cement class G mechanical parameters
inventory corresponding to curing condition was an obstacle for them to carry on with their
20 |
ADE | modelling approaches. Considering that, the mechanical properties of the cement are significantly
dependent on the curing conditions, which vary along its depth and its exposure to formation fluids
[20, 88, 89]. Subsequently, many laboratory tests have been carried out on well cement to
determine the key parameters for modelling purposes reviewed in the following sections.
Cement Mechanical Properties
Thiercelin, Dargaud, Baret and Rodriquez [63] presented an experimental study utilising
cement class G with varieties of additives to determine the material’s flexural and compression
strength, and Young's modulus in flexion and compression. The tensile properties were obtained
using three-point bending tests on 30×30×120 mm prisms, with a loading rate of 0.01 cm/min.
The compressive properties were measured via uniaxial compression tests on 50.8×50.8×50.8 mm
(2×2×2 in) cubes. The volume of additives, curing conditions, and slurry density were different
for each test. Therefore, it is difficult to associate any differences in mechanical properties with
one specific factor.
Bosma, Ravi, van Driel and Schreppers [44] carried out confined and unconfined compression
tests to obtain cement mechanical properties for their modelling. However, they did not publish
the experimental details and the outcomes.
Reddy, Santra, McMechan, Gray, Brenneis and Dunn [90] used acoustic measurements to
compare dynamic cement mechanical properties with static mechanical properties. They used
cylindrical samples with the size of 50.8×102 mm (2×4 in) and cured them under pressure of 20.7
MPa (3000 psi) and temperature of 88oC for 72 hrs in an autoclave and cooled down to room
temperature and also depressurized slowly. They performed confined and unconfined compression
tests. However, the results of confined compression tests haven’t been published. Their
observations showed the importance of the time period to achieve long-term mechanical properties
and found a correlation between static and dynamic mechanical properties, i.e. dynamic modulus
values were 1.6 times higher than the static values.
Roy-Delage, Baumgarte, Thiercelin and Vidick [91] planned a slurry formulation with cement
class G to achieve highly durable cement. They cured the samples at 77oC and 114oC with a
pressure of 20.7 MPa (3000 psi) for three days or upon reaching a constant compressive strength.
Three-point bending tests and crushing tests were performed on 30×30×120 mm and 50.8× 50.8×
50.8 mm (2× 2× 2 in) cubes, respectively. This study showed that cement cured at the higher
temperature (114oC) has a higher Young's modulus and uniaxial compressive strength but a lower
Modulus of rupture in flexure.
21 |
ADE | Nasvi, Ranjith and Sanjayan [92] used cylindrical samples with the size of 50 ×100 mm to
measure the uniaxial compression strength of cement class G. The samples were oven cured at
different temperatures between 300C to 800C for 24 hours excluding the samples required to be
cured at room temperature. Afterwards, all of the samples were kept at ambient temperature for
another 48 hours. Their results demonstrated that samples that cured at 600C had the maximum
uniaxial compressive strength of 53 MPa, but that samples cured above this temperature presented
a lower uniaxial compressive strength. The Young’s modulus of cement class G is higher at lower
curing temperatures and reaches its maximum value at the curing temperature of 400 𝐶.
James and Boukhelifa [93] provided a comprehensive review on the published experimental
studies and recommended a set of measurements methods to determine cement mechanical
parameters (Young’s modulus, Poisson’s ratio, unconfined compressive stress (UCS), and tensile
strength) as inputs for wellbore integrity models and they validated their approach by filed
evaluation at actual wells. They suggested using suitable load frame equipment with controllable
load and displacement rates. Based on their results Young’s modulus and the Poisson’s ratio are
independent of confining stress. They believed that Brazilian tests estimate the cement tensile
strength 50%-75% higher than the actual value. Therefore, the commonly employed rule-of-thumb
estimation for tensile strength (tensile strength= 10% of UCS) substantially adds more safety
factor to the estimations.
Guner and Ozturk [94] measured both uniaxial compressive strength and Young's modulus at
different cement curing periods of 2, 7, and 14 days. They concluded increasing the curing time
enhances the mechanical properties of cement by 2-3 times.
Teodoriu and Asamba [95] investigated the effect of salt concentration on cement class G
properties by performing uniaxial compression tests on cubic samples with the size of 50.8× 50.8×
50.8 mm (2× 2× 2 in). They cured samples in water in atmospheric condition for 24hrs and then
was placed in an autoclave for curing period of one to seven days under two different conditions
(30oC and 10 MPa / 150oC and 20 MPa). They also measured the compressive strength of samples
cured at the atmospheric condition at the age of 21 days. Their results of the batch without salt
(0% BWOW-By weight of Water) with respect to the first curing condition were summarised in
Table 3.1 for comparison purposes. They showed the samples with 5% ± 2.5% salt concentration
curing at atmosphere to moderate temperature yield the maximum compressive strength among
all the other samples with different salt concentration curing at different conditions.
Romanowski, Ichim and Teodoriu [96] compared two methods for measuring the cement
compressive strength (ultrasonic pulse velocity versus mechanical method). The tests were
22 |
ADE | performed at different curing times on cement class G, cement class G with bentonite, and cement
class G with other additives. They demonstrated that the outcomes of ultrasonic methods should
be calibrated using the mechanical (destructive) measuring methods. The importance of achieving
an extensive database on wellbore cement was emphasised as well.
Latest studies in oil and gas cementing technology emphasis that cement sheath mechanical
failure occurs not only because of imposed compressive stresses but also because of tensile
stresses [97]. However, there are no API guidelines for measuring the tensile properties of cement,
and ASTM standards for the measurement of tensile properties of concrete present various
weaknesses when applied to cement tensile tests. As, these standards have been designed for
cement at locations only a few meters down the ground and they don’t incorporate the curing
conditions with respect to high pressure and high temperature in harsh conditions, i.e. downhole
conditions [97, 98].
Heinold, Dillenbeck and Rogers [97] cured samples made of cement class G plus additives in a
standard high-pressure high-temperature (HPHT) curing chamber, under a pressure of 20.7 MPa
at two different temperatures of 37.8oC and 93.3oC for 72 hours. They performed uniaxial
compression tests, flexural strength tests, and tensile strength tests on 50.8×50.8×50.8 mm (2×2×2
in) cubes., 40.6×40.6×160.02 mm (1.6×1.6×6.3 in) prismatic specimens, and dog bone specimens,
respectively. They showed that the correlation between unconfined compression strength and
tensile strength (empirical relations) does not always apply. According to their results, samples
cured at a higher temperature (93.3oC) demonstrated lower flexural strength and higher tensile
strength. Heinold, Dillenbeck, Bray and Rogers [99] continued this study by curing samples at
two different temperatures of 54.40C or 82.20C for 48 hours in an atmospheric water bath. The
authors compared the results of splitting tensile strength (STS) tests with direct tensile tests on the
dog-bone sample. The splitting tensile strength test results overestimated the tensile properties of
cement class G by order of 1.5 to 2.5. However, direct tensile test measurements were also
impacted by stress concentrations on the samples at or near grip points, which may cause the
premature breakage of the samples.
Dillenbeck, Boncan, Clemente and Rogers [98] performed uniaxial tensile tests on dog-bone
samples made of cement class H and additives to measure its uniaxial tensile strength. They
developed a new testing machine to simulate downhole conditions in a wellbore during curing and
performed tensile tests on dog-bone samples. The results showed that the uniaxial tensile strength
of the cement samples was highly dependent on the stress loading rate. Therefore, the authors
23 |
ADE | Labibzadeh, Zahabizadeh and 38 2.8 2 14.24
Khajehdezfuly [104]
Labibzadeh, Zahabizadeh and 68 17.2 2 12.72
Khajehdezfuly [104]
Labibzadeh, Zahabizadeh and 82 41.4 2 18.82
Khajehdezfuly [104]
Labibzadeh, Zahabizadeh and 121 51.7 2 16.4
Khajehdezfuly [104]
Labibzadeh, Zahabizadeh and 149 51.7 2 4.59
Khajehdezfuly [104]
30 10 1 2
Teodoriu and Asamba [95]
30 10 3 18
Teodoriu and Asamba [95]
30 10 7 28
Teodoriu and Asamba [95]
ambient ambient 21 42
Teodoriu and Asamba [95]
As can be seen in Table 3.1, the cement class G mechanical inventory lacks triaxial compressive
properties in general and uniaxial compressive properties corresponding to the different curing
temperatures, particularly in long-term curing periods.
It is also should be noted that the pre-peak and post-peak behaviour in the stress-strain graphs
vary according to different specimen size and shape [105]. Table 3.1 shows the majority of studies
were performed on cubic samples. While cylindrical specimens might be more suitable to be
employed since cube tests provide higher values (the uniaxial strength measured using sufficiently
slender specimens is usually around 70%-90% of the cube strength [106]). In cubic samples, the
restraining effect of the platens spreads over the total height of a specimen, but in cylindrical
samples, some parts of specimens stay unaffected [107]. Another problem regarding experiments
using cubes is that the post-peak behaviour is milder, therefore, requires more energy consumption
than using cylinders. The effect of size specimen is also larger for cubic samples [105].
Consequently, the experimental aspect of this thesis (Chapter 6), aims to add the discussed
missing values to the cement class G mechanical inventory; the uniaxial and triaxial compression
tests were performed on cylindrical samples cured at two different curing temperatures.
26 |
ADE | regards to high pressure and high temperature in harsh conditions, i.e. downhole conditions [97,
98].
Table 3.2 demonstrates that the measurement of tensile strength and particularly the fracture
energy of cement class G, in particular over long-term periods, were simply overlooked in many
experimental studies.
Subsequently, in the experimental aspect of this study, three-point bending tests were performed
on notched and un-notched beams to measure the tensile strength and fracture energy. The splitting
tests and direct tensile tests were avoided due to their overestimated results and stress
concentrations issue on the direct test samples at or near grip points (point loading), which
provides high potential for immature breakage of the samples [97].
Cement Thermal Properties
Bentz [109] measured heat capacity and thermal conductivity of hydrated cement paste with
the help of a transient plane source method as a function of water to cement ratio, curing condition,
and degree of hydration. Samples were cured under sealed and saturated conditions. The pattern
for both is almost the same. In the early stages of hydration, heat capacity was decreased and then
stabilized to a constant value. Additionally, it can be understood that there is a little deviation in
the thermal conductivity of samples due to the degree of hydration and curing condition.
Because of the enhanced oil recovery process, i.e. steam-assisted gravity drainage or cyclic
steam stimulation, the temperature of wells can raise to 3500 C [95]. Considering this situation,
the linear coefficient of thermal expansion (LCTE) is an important factor to simulate the coupling
effect of thermal and mechanical effects on the well integrity failure. Based on this factor the
position of the cement-failure zone and cement integrity zone can be varied [110]. According to
Loiseau [110], LCTE is a function of chemical components and oil well cement curing
temperature. Additionally, the LCTE becomes more of an influential factor as the temperature of
the bottom hole of wells changes drastically. They stated some precautions to measure LCTE
precisely and also investigate the effects of chemical compositions and temperature on LCTE. The
μ
LCTE of net cement was measured as 9.10 ⁄oC .
Nygaard, Salehi, Weideman and Lavoie [74] measured the cement thermal conductivity using
divided-bar apparatus and the cement specific heat capacity using steel-frame equipment consisted
of one LVDT and a glass beaker filled with water.
Considering the limitations in the literature review, it would be more realistic to perform the
three-dimensional simulations to be able to predict the fracture initiation and propagation in
28 |
ADE | 4. Research Objectives
The overall objective of this study is to improve the modelling capabilities to assess cement
sheath integrity by employing a more suitable constitutive model for the cement sheath. The
experimental outcomes on the behaviour of the cement-based specimens under compression tests
showed a strong non-linearity in the obtained stress-strain curve which confirms the necessity of
applying plasticity theories. However, it is difficult to describe the elastic stiffness degradation of
the cement-based materials which occurs during experiments by using the classical plasticity
theories. Therefore, in this study, Concrete Damage Plasticity (CDP) model developed by [54]
and then modified by [55] has been applied. The Concrete Damage Plasticity model combines
plasticity and damage mechanisms and is particularly formulated for modelling geo-materials such
as rocks, concrete, and cementitious materials by considering two different tensile and
compressive damage variables. Considering that, the response of geo-materials in compression
and tension is very different. The yield criterion in the CDP model also represents the pressure-
dependency of the materials behaviour under shearing at different levels of confinement in
addition to the incorporation of non-associated flow rule (material dilatancy). Based on the above
discussion, the CDP model can be considered a very suitable model for the wells cement under
different loading conditions in wellbore integrity assessments.
However, the incompleteness of cement class G mechanical parameters inventory, e.g. lack of
experimental data under different confining pressure and tensile properties, was an obstacle to
perform accurate simulations. For instance, the confined (triaxial) compression tests data are
required to approximate the shape of the shape of yield / and failure surfaces. The measurement
procedures of cement tensile strength properties and fracture energy required to simulate the
tensile behaviour mechanism are not consistent. Moreover, the effect of curing temperature on the
cement long-term mechanical properties was also missing from the literature.
Therefore, this study aims to add these cement mechanical properties to the cement class G
inventory as well. The data and analyses added to the cement class G inventory are as follows.
The investigations of the curing temperature and pressure confinements effects on the strength
and post-peak response of the cement class G under compression. In addition, in the absence of
API guidelines for measuring the cement tensile properties, the methods for measuring cement
tensile and / flexural strength were not consistent, and the measurement of cement fracture energy
was mostly overlooked.
As a final point, this research leads to better cement sheath integrity evaluations subjected to
pressure and temperature variations using the CDP model along with the incorporation of
31 |
ADE | calibrated constitutive parameters obtained from the experimental data. The connections between
the research aims and the associated publications are discussed as follows.
4.1. Objective 1:
Evaluation of Cement Sheath Integrity Subject to Enhanced Pressure (Paper-1)
The cement sheath should be designed and placed in a way to resist the imposed stresses from
wellbore operational procedures. In this paper, the effect of elevated pressure on the cement sheath
integrity was investigated.
The key objective of this paper was the incorporation of a suitable constitutive model (Concrete
Damage Plasticity) for the cement sheath in wellbores. The CDP model especially formulated for
modelling geo-materials including rocks, and cement-based materials by considering the
difference in tensile and compressive material responses, the pressure-dependent material
behaviour under shearing at different levels of confinement, and the martial dilatancy. Therefore,
it is advantageous in terms of its capabilities to predict both cement sheath compression/shear and
tensile mechanical failure in wellbores.
To determine the corresponding constitutive model parameters, uniaxial compression tests,
three-point bending tests on specimens manufactured from class G well cement, and experimental
data calibration and validation should be performed.
The integrity of the cement sheath should be assessed in relation to the local and global
compression (crushing) or tensile (cracking) damage indicators within the cement sheath.
The interfaces should be properly modelled to investigate the effect of elevated pressure on the
creation of micro-annulus (de-bonding occurrence) as well. Thus, the surface-based cohesive
approach was employed.
To investigate the effect of different influential factors including the orientation and the
anisotropy of in-situ stresses, and different stiffness’s of rocks on the cement sheath integrity,
sensitivity analyses should be performed.
4.2. Objective 2:
Effect of Curing Conditions on the Mechanical Properties of Cement Class G with the
Application to Wellbore Integrity (Paper-2)
The paucity of experimental data is an obstacle for cement sheath integrity modelling, for
instance, to determine the shape of yield / and failure surfaces the experimental data obtained from
confined (triaxial tests) over a wide range of confinement are required. Additionally, the cement
tensile properties including the tensile/flexural strength and fracture energy are required to
32 |
ADE | simulate the tensile behaviour mechanism. However, in the absence of API guidelines for
measuring the cement tensile properties, the methods for measuring cement tensile and / flexural
strength were not consistent, and the measurement of cement fracture energy was mostly ignored.
The effect of curing temperature on the cement long-term mechanical properties was also missing
from the available literature. This paper aims to fill these gaps and expand the cement class G
mechanical inventory.
The specimens made out of cement class G cured at two different temperatures (30o C and
70o C). The unconfined (uniaxial), confined (triaxial) compression tests and three-point bending
tests were performed to measure the corresponding cement mechanical properties.
The suitability and reliability of the intended parameters should be calibrated and validated by
the numerical simulations of experiments.
4.3. Objective 3:
Evaluation of Cement Sheath Integrity Reflecting Thermo-Plastic Behaviour of the Cement
Sheath in Downhole Conditions (Paper-3)
The objective of this study was to investigate the effect of pressure and temperature variations
on the cement sheath integrity employing CDP model along with the determined corresponding
constitutive parameters from the previous experimental studies for the cement sheath.
The integrity of the cement sheath should be assessed in relation to the creation of compression
(crushing) / tensile (cracking) damage within the cement sheath considering different thermal-
mechanical loading scenarios.
The interfaces should be modelled in a way to convey the traction-separation capacity (cohesive
behaviour and damage evolution law) and thermal conduction simultaneously.
The effect of different heating/cooling rates along with different wellbore architectures and
different degrees of eccentricities should be investigated to reflect the realistic situations in
wellbores.
The connections between the research aims and associated publication are summarised in
Figure 4.1.
33 |
ADE | Evaluation of Cement Sheath Integrity Subject to Enhanced Pressure
(PAPER-1)
ABSTRACT
Well-cementing (cementation) is an influential stage of a wellbore completion, as the cement
sheath is responsible for providing complete zonal isolation. Therefore, it is of utmost importance
to understand the cement mechanical failure mechanisms since well cement failure and interfacial
debonding between the cement and casing and cement and rock formations can lead to a barrier
failure. During the wellbore lifetime, a cement sheath is subjected to pressure loading variations.
This paper demonstrates the results of an experimental-numerical study to investigate the cement
sheath integrity after being subjected to an enhanced pressure. A constitutive model specifically
formulated for the modelling of quasi-brittle materials is applied to the investigation of cement
sheath integrity, incorporating both compression and tensile damage mechanisms. Laboratory
experiments are carried out to obtain strength properties of cement class G followed by calibration
of the model parameters based on the obtained experimental results. A three-dimensional finite
element framework employing the constitutive model for cement sheath and a surface-based
cohesive behaviour for the interfaces is developed for integrity investigations. The effects of
different orientations of in-situ stresses, different stiffness’s of surrounding rock, and the
eccentricity of the casing within the wellbore on the integrity of the cement and interfaces are
investigated. The significance of cement sheath centralisation and elevated risk of cement
mechanical failure caused by wellbore operations in anisotropic fields with soft rocks formation
were highlighted. Furthermore, the relatively high magnitude of tensile damage (cracking index)
within the cement sheath confirms the importance of tensile properties to be incorporated into the
constitutive modelling.
Keywords: cement sheath integrity, concrete damage plasticity model, casing eccentricity,
anisotropic in-situ stresses, compression damage, tensile damage
5.1. Introduction
Four million onshore hydrocarbon wells have been drilled worldwide [9], with nearly 10000 in
Australia alone (from data retrieved from Geoscience Australia) [9]. The cement placed in the
annular gaps between casing strings and the formation is a key barrier to provide zonal isolation
and maintain the integrity of the wellbore [111]. The integrity of the annular cement and cement
interfaces has the potential to be compromised in each of the wellbore operations, including but
not limited to, continuous drilling operations, completion operations, stimulation treatments,
pressure integrity testing (PIT), and production processes [112]. Therefore, understanding of
37 |
ADE | cement failure mechanisms under different operating conditions is of the utmost importance for
better assessment of wellbore integrity.
Failure of the cement sheath within a wellbore is affected and governed by material mechanical
properties (cement compressive strength [31, 44, 45], Young’s modulus [31, 44, 45], tensile
strength [46], and bond strength [47]), loading conditions (in-situ stresses [44, 46, 48]), cement
history (cement shrinkage) [8], and also wellbore architecture (cement sheath diameter, formation
properties, cement sheath eccentricity, and wellbore deviation [8, 37]).
Mechanical integrity models investigated to this point can be categorised into analytical models
and numerical models. Analytical methods are generally performed by applying simplified
assumptions to facilitate finding solutions. The accuracy of analytical models and subsequently
their solutions are limited to the correctness and suitability of their initial assumptions and
simplifications. Thiercelin, Baumgarte and Guillot [113] modelled the stress state within the
cement sheath assuming the linear-elastic properties for cement, axisymmetric geometry, and fully
bonded or unbound situations for the interfaces. Shi, Li, Guo, Guan and Li [65] estimated the
initial radial and tangential stresses at cementing interfaces with the assumption of axisymmetric
geometry, isotropic horizontal in-situ stresses and elastic properties for the cement sheath and
interfaces. Honglin, Zhang, Shi and Xiong [64] proposed a model using Mohr-Coulomb criterion
and multi-layer thick wall theory assuming plane strain conditions, and all the wellbore
components are deemed as thick-walled cylinders and completely bonded. However, some of
these assumptions and simplifications may lead to unrealistic results. For instance, failure modes
in all directions would not be captured in two-dimensional models (plane strain), and the
axisymmetric geometry and assumed isotropic in-situ stresses do not correctly reflect the real
conditions.
Numerical modelling can be very advantageous regarding its ability to incorporate material
non-linearity, different types of geometry and boundary conditions, and in-situ stress conditions
[47]. The accuracy of these numerical models is reliant on the validation and verification of
experimental data.
Nabipour, Joodi and Sarmadivaleh [28] simulated downhole stresses using FEM along with
sensitivity analysis on casing internal pressure, anisotropic horizontal in-situ stresses, and casing
eccentricity. They have used a plane strain model with thermo-elastic material properties, and the
interfaces are assumed to be fully bonded. Wang and Taleghani [37] performed three-dimensional
(3-D) poroelastic simulations to assess the integrity of the cement sheath around wellbores. The
interfaces have been modelled using porous cohesive elements. The cohesive parameters were
38 |
ADE | determined by running inverse analyses on the bonding studies carried out by Evans and Carter
[86]. Despite, the massive progress regarding interface modelling, the use of elastic behaviour for
cement sheath is an over-simplification that can affect the accuracy and reliability of the results.
Fleckenstein, Eustes and Miller [51] employing von-Mises criteria, they demonstrated that the
magnitude of tangential stresses would be greatly decreased if the cement sheath acts as a ductile
material with lower Young’s modulus and higher Poisson’s ratio which is in agreement with
Goodwin and Crook [32]. The lack of pressure dependency of the von Mises criteria is however
problematic in modelling cementitious materials. To overcome this shortcoming, a number of
researchers have adopted the Mohr-Coulomb criteria in their work.
Bosma, Ravi, van Driel and Schreppers [44] used a two-dimensional model considering
symmetry geometry for the wellbore. Mohr-Coulomb plasticity combined with smeared cracking
description was used to model the cement sheath under compression / shear and tension. The
cement sheath interfaces were modelled using interface elements applying a coulomb friction
criterion. Nygaard, Salehi, Weideman and Lavoie [74] performed an experimental-numerical
study using Mohr-Coulomb plasticity model for the cement and formation to investigate the effect
of dynamic loading on wellbore leakage. Their parametric study showed that cement with higher
Young’s modulus and Poisson’s ratio are detrimental factors causing radial fractures, tensile
failure and debonding. However, utilising cement with low strength mechanical properties
increases the risk of shear failure within the cement sheath.
The Mohr-Coulomb model assumes a linear relationship between √𝐽 and 𝐼 in the meridian
2 1
plane, while this relationship has been experimentally shown to be non-linear [76, 77], for
cementitious materials, particularly at low confinement. The major principal stress 𝜎 and
1
intermediate principal stress 𝜎 are defined independently in Mohr-Coulomb model which results
2
in underestimation of the yield strength of the material and, it is not in a good agreement with
experiments in which the effect of 𝜎 is being considered. The shape of yield surface in the
2
deviatoric plane is an asymmetrical hexagon, whereby the sharp corners can hinder convergence
in numerical simulations [70, 78]. Moreover, quasi-brittle materials experience a huge volume
change due to a large amount of inelastic strains (dilatancy) which has been overlooked so far by
using associated flow rules in the modelling of the cement. The associative plastic flow rules tend
to lead to poor results in dilatancy evolution [55].
The use of the modified Cam-Clay model has been suggested as a method to incorporate cement
micro cracking mechanisms by Bois, Garnier, Rodot, Sain-Marc and Aimard [36] owing to the
nonlinearity of stress-strain curve achieved from the isotropic drained compression tests [72] and
39 |
ADE | heterogeneous nature of cement at the microscale. Although important aspects of materials
behaviour (material strength, compression or dilatancy, and critical state of elements under high
distortion) are considered in this model, the tensile post-peak material is not incorporated into this
framework.
Numerical modelling has been significantly improved regarding complexity and ability to
model wellbore integrity assessment with a high degree of accuracy. The incorporation of
appropriate material constitutive law, particularly with regards to cracking behaviour, and
consequently the evolution of corresponding constitutive parameters still requires attention.
Bosma, Ravi, van Driel and Schreppers [44] advocated the used of smeared cracking models in
combination with plasticity and Salehi [114] have employed a discrete crack methodology via the
use of nonlinear fracture mechanics for cohesive cracks. Therefore, in this study, the concrete
damage plasticity (CDP) model [54, 115] was used to investigate cement mechanical failure. This
model incorporates a non-associative flow rule and damage under both tensile and compressive
stress states, which is more appropriate for the characterisation of cementitious materials.
This paper is organised as follows; Section 5.2 describes cement constitutive modelling
including the experimental procedures to achieve mechanical properties, the concrete damage
plasticity model as the appropriate constitutive model to be utilised, and the calibration of the
model parameters according to the performed experiments. Surface-based cohesive behaviour is
introduced for interface modelling and followed by determination of cohesive model parameters
in section 5.3. Section 5.4 describes finite element modelling including model components,
material properties, geometry and discretisation, initial and boundary conditions. Section 5.5
describes the results of cement sheath and the interfaces integrity investigations for the different
initial state of in-situ stresses followed by conclusion in section 5.6.
5.2. Cement Constitutive Modelling
Portland Class G (API rating) well cement is predominantly utilised as the basis of well cement
blends [34], additives are incorporated to obtain certain properties such as enhanced strength or
reduced weight [34]. In general, the permeability of cement used in oil and gas industry (cement
class G) is very low usually less than 0.1 mDarcy [11]. Therefore, hydraulic isolation will be
achieved, and any probable leakage pathways can be created only through flaws resulting from
issues in cement placement procedures or mechanical failure due to the variation of pressure
during wellbore operations.
40 |
ADE | Experimental Procedures
The concrete damage plasticity model (CDP) has been calibrated and verified according to the
experiments have been performed by Arjomand, Bennett and Nguyen [116]. The specimens were
made of cement class G cured at 300C for 28 days in a pre-heated water tank with a manageable
thermostat. The slurry density was 1.9 g/cc corresponding to water to cement mass ratio of 0.44.
Prior to testing, the surfaces of samples were ground to obtain smooth ends, so the ends were
perfectly orthogonal to the longitudinal cylinder axis [77].
In this study, relatively slender cylindrical specimens were employed to avoid problems with
platen restraint that are encountered using squat cube specimens [35]. The uniaxial strength
measured using sufficiently slender specimens is usually around 70%-90% of the cube strength
[106]. The uniaxial compressive strength was determined using 42 mm diameter, 100 mm long
cylindrical specimens which deliver aspect ratio of 2.4. It also helps to minimise the effect of
specimen shape on the determination of the modulus of elasticity [117].
To investigate the effect of displacement rate on the cement uniaxial compressive behaviour
three displacement rates of 0.2 mm/min, 0.1 mm/min and 0.04 mm/min were investigated. The
samples showed highly brittle behaviour at displacement rates of 0.2 mm/min and 0.1 mm/min.
To capture post-peak behaviour, the displacement rate was reduced to 0.04 mm/min at which the
specimen displayed less brittle behaviour.
Figure 5.1 demonstrates the uniaxial compressive behaviour of the specimens were subjected
to a constant displacement rate of 0.04 mm/min. The axial displacement of the loading platen was
measured by two external 25 mm span linear variables differential transformers (LVDT) were
installed at the bottom platen on the sides of the specimen. The obtained results are in a good
agreement a subset of the data detailed in Teodoriu, Asamba and Ichim [118], Teodoriu, Amani,
Yuan, Schubert and Kosinowski [119]’s for compressive strength of class G without additives at
the age of 28 days.
41 |
ADE | Figure 5.1: Cement Class G Compressive and Tensile Response Respectively
To determine the tensile stress of the cement according to ASTM standard C348-02 “Standard
Test Method for Flexural Strength of Hydraulic-Cement Mortars” [120] three-point bending tests
were run on beams with dimensions of 160×40×40 mm. The suggested loading rate by the standard
was 2640±110 N (600±25 lbf/min) was very fast. Therefore, the tests were performed by applying
displacement rate of 0.015mm/min. The axial displacement of the loading platen was measured
via using of two LVDTs installed on both sides of the beam specimens. The corresponding load-
displacement is shown in Figure 5.1. The tensile strength 𝜎 for prisms was calculated, from the
𝑡
bending tests as:
3𝐹𝑠
𝜎 = (5.1 )
𝑡 2𝑏𝑑2
where F is the applied load, s in the span of the beam, b and d are width and depth of the specimen
respectively.
Concrete Damage Plasticity Model Description
A continuum model based on damage mechanics and plasticity theory can be used to better
describe the behaviour of class G cement, from initial yield to failure. In this study, we employ
42 |
ADE | the concrete damage plasticity (CDP) model initially developed by Lubliner et al. (1989) and
expanded by Lee and Fenves (1998). This model includes two damage variables for tensile and
compressive failure, taking into account unilateral effects. The elastoplastic behaviour is
decoupled from degradation damage response which leads to easier implementation [54, 55, 115,
121].
The uniaxial tension response is characterised by a linear elastic relationship until reaching the
failure stress (𝜎 ) which corresponds to the beginning of micro-cracking in the material as
𝑡
calculated based on equation 5.1 and 5.2. Beyond the failure stress, the effects of micro-cracking
is taken into account in the model using a softening stress-strain response. The uniaxial
compression response is also characterised by a linear elastic relationship until reaching the initial
compressive strength (𝜎 ) followed by stress hardening in the plastic region up to the ultimate
𝑐
stress (𝜎 ). Strain softening occurs subsequent to reaching the ultimate stress.
𝑐𝑢
The stress-strain relations under uniaxial tension and compression are defined as follows
respectively.
𝜎 = (1−𝑑 )𝐸 (𝜀 −𝜀̃𝑡 ) (5.2)
𝑡 𝑡 0 𝑡 𝑝𝑙
𝜎 = (1−𝑑 )𝐸 (𝜀 −𝜀̃𝑐 ) (5.3)
𝑐 𝑐 0 𝑐 𝑝𝑙
where 𝑑 and 𝑑 are tensile and compression damage variables; 𝐸 is initial undamaged
𝑡 𝑐 0
stiffness; 𝜀̃𝑡 , 𝜀̃𝑐 are tensile and compressive equivalent plastic strains respectively.
𝑝𝑙 𝑝𝑙
The shape of yield surface in the deviatoric plane changes according to the ratio of the second
stress invariant on the tensile meridian to the compressive meridian which allows better capture
of material behaviour. This yield function was defined by Lubliner, Oliver, Oller and Onate [54]
with some modifications made by Lee and Fenves [115] afterwards to interpret the evolution of
strength under tension and compression. It is defined as follows.
1
𝐹 = 1−𝛼(𝑞̅−3.𝛼.𝑝̅+𝛽(𝜀 𝑝̃ 𝑙)〈𝜎̅ 𝑚𝑎𝑥〉−𝛾〈−𝜎̅ 𝑚𝑎𝑥〉)−𝜎̅ 𝑐(𝜀 𝑝̃𝑐 𝑙)= 0 (5.4)
1
where 〈 〉 is the Macaulay bracket, 𝜎̅ is the maximum principle effective stress, 𝑝̅ = − 𝜎̅ .𝐼
𝑚𝑎𝑥
3
is the effective hydrostatic stress and 𝑞̅ = √3 𝑆̅.𝑆̅ is the Mises equivalent effective stress with 𝑆̅ =
2
𝑝̅𝐼+𝜎̅ being the deviatoric part of the effective stress tensor. The function 𝛽(𝜀̃ ) in (5.4) is
𝑝𝑙
defined as follows.
𝜎̅ (𝜀̃𝑐 )
𝑐 𝑝𝑙
𝛽(𝜀̃ )= (1−𝛼)−(1+𝛼) (5.5)
𝑝𝑙 𝜎̅ (𝜀̃𝑡 )
𝑡 𝑝𝑙
in which two cohesion stresses are employed for the modelling of cyclic behaviour.
43 |
ADE | 𝜎
( 𝑏𝑜)−1
𝜎
𝛼 = 𝑐 (5.6)
𝜎
2( 𝑏𝑜)−1
𝜎
𝑐
𝜎
where 𝑏𝑜 is the ratio of biaxial compressive yield stress to uniaxial compressive yield stress.
𝜎𝑐
𝜎
Experimental values used for concretes for 𝑏𝑜 vary between 1.10 and 1.16 which result in
𝜎𝑐
parameter 𝛼 in the range of 0.08 ≤ 𝛼 ≤ 0.1212 [54, 122].
The shape of loading surface in the deviatoric plane is controlled by parameter 𝛾 in Equation
(5.4) [123] and define as
3(1−𝐾 )
𝑐
𝛾 = (5.7)
2𝐾 +3
𝑐
where 𝐾 =
(√𝐽2)𝑇𝑀
is a coefficient determined at a given state 𝑝̅ , 𝐽 is the second invariant of
𝑐 2
(√𝐽2)𝐶𝑀
stress with the subscripts TM and CM employed for the tensile and compressive meridians
respectively and must satisfy the condition 0.5 ≤ 𝐾 ≤ 1 . Typical values of 𝐾 for concrete have
𝑐 𝑐
been suggested from 0.64 by Schickert and Winkler [124] and 0.66 by Richart, Brandtzaeg and
Brown [125] to 0.8 by Mills and Zimmerman [126]. Lubliner, Oliver, Oller and Onate [54] used
𝐾 = 2/3 for plain concrete which results in γ=3.
𝑐
For the non-associated flow rule, the plastic potential 𝐺 in the form of the Drucker-Prager
hyperbolic function is used.
𝐺 =√(𝜖𝜎 tan𝜓)2+𝑞̅2−𝑝̅.tan𝜓 (5.8)
𝑡𝑜
In which 𝜎 is the uniaxial tensile stress at failure, the dilation angle 𝜓 is measured in a p-q
𝑡𝑜
plane at high confining pressure, and 𝜖 is an indicator for the eccentricity of the plastic potential
surface.
Calibration of Concrete Damage Plasticity Parameters for Cement Class G
Determination of constitutive parameters is significantly important in FE analysis to minimise
the error of the models in the analyses [127]. The constitutive parameters have to be calibrated in
a way to have a good connection with experimental data [128]. To calibrate the corresponding
parameters in the concrete damage plasticity model for cement class G, the values for cement
Young’s modulus 𝐸 , cement initial compressive stress 𝜎 , ultimate compressive stress 𝜎 , and
0 𝑐 𝑐𝑢
tensile strength 𝜎 were extracted from the uniaxial compression tests and three-point bending
𝑡𝑜
tests respectively detailed in section 5.2.1.
44 |
ADE | Some data pertaining to the confinement dependent strength of well cements is available in the
open literature, for example [129, 130]. However, the biaxial to uniaxial strength ratio required
for the characterisation Lubliner, Oliver, Oller and Onate [54] plasticity model is difficult to
extract from triaxial data. In addition, the post-peak material behaviour is less well reported, even
for simple stress states. Therefore, the
𝜎
𝑏𝑜⁄𝜎 , 𝛾 , 𝜓, and 𝜖 parameters are calibrated for cement
𝑐
class G in this section.
A three-dimensional uniaxial compressive test was simulated in ABAQUS/standard to find the
best match between the performed uniaxial compression experiment test in the laboratory and
numerical one. The geometry and the boundary conditions are depicted in Figure 5.2. A finite
element mesh was considered for the simulation purposes consisting of 42456 8-noded hexahedral
elements. The simulations were performed using a reasonable range for each parameter to obtain
the best fit to experimental data. The trial range of dilation angle 𝜓 was between 250 to 450, the
eccentricity 𝜖 was examined between a range of 0.01 to 0.1, the ratio of biaxial compressive yield
stress to uniaxial compressive yield stress was tried between 1.1 to 1.17, and the 𝐾 which is the
𝑐
ratio of √J in tensile meridian to compressive meridian were tried from 0.5 to 1.
2
The closest match to the experimental results found by using dilation angle 𝜓=420,
eccentricity 𝜖=0.1,
𝜎𝑏𝑜
=1.16, and 𝐾 =0.8. The results and corresponding failure patterns in the
𝑐
𝜎𝑐
laboratory and ABAQUS are shown in Figures 5.2 and 5.3. It can be seen that the simulation
results can match the experimental counterparts in terms of both macro responses and failure
pattern, indicating that the calibrated set of parameters are appropriate for the modelling of this
class G cement.
45 |
ADE | Table 5.1: Cement Class G Mechanical Properties Obtained from the Experiments and Calibration
Process
Young’s modulus Dilation angle Fracture energy
𝑬 (GPa) 𝝍 (degrees) 𝑮 (N/mm)
𝟎 𝒇
6.8 42 35
Inelastic
Eccentricity 𝑲 Initial compressive
𝒄
strain
𝜺 stress 𝝈 (MPa)
𝒄 𝜺̃𝒄
𝒊𝒏
0.1 0.8 50 0.007353
Ultimate Tensile stress
Cracking Strain
compressive stress 𝝈 (MPa)
𝒕 𝜺̃𝒕
𝒄𝒌
𝝈 (MPa)
𝒄𝒖
55 1.92 0.0000485
This calibration procedure has enabled us to identify the remaining model parameters required,
however there remain large uncertainties in applying these parameters to real world scenarios. The
cement curing (duration, temperature, pressure), the cement mix design, operating temperature
and stress history that a wellbore has experienced will all effect the state of the material.
5.3. Interface Modelling
The interfaces of the cement with the casing and the formation are recognised as the weakest
link in providing an effective barrier to leakage [37, 81]. Its behaviour and failure can be described
by a cohesive model for interfaces between two different materials [81]. In this study, the cement
sheath interfaces are represented by surface-based cohesive behaviour defined as surface
interaction property with traction transferring capacity. The relationship between tractions 𝑡 and
separation 𝛿 can be described using a traction – separation law (Equation 5.10).
𝑡 𝐾 𝐾 𝐾 𝛿
𝑛 𝑛𝑛 𝑛𝑠 𝑛𝑡 𝑛
{𝑡} = {𝑡 }= [𝐾 𝐾 𝐾 ]{𝛿 }= 𝐾{𝛿} (5.10)
𝑠 𝑠𝑛 𝑠𝑠 𝑠𝑡 𝑠
𝑡 𝐾 𝐾 𝐾 𝛿
𝑡 𝑡𝑛 𝑡𝑠 𝑡𝑡 𝑡
where the subscripts n, s, t refer to the normal and shear directions along the interface. K are
stiffness components, which are coupled in all directions.
47 |
ADE | Figure 5.6: Linear Softening Traction-Separation Law
The cohesive constraint is enforced at each slave node for cohesive surfaces. Contact separation
is expressed as the relative displacements between the slave surface nodes and their matching
opposite nodes on the master surfaces along the contact normal and shear directions. Stresses are
defined for the surface-based cohesive surfaces as the cohesive forces acting along the contact
normal and shear directions divided by the contact area at each contact point.
The degradation and eventual failure of the bond between two cohesive surfaces are described
by a damage law. The damage mechanism is defined based on damage initiation criterion and
damage evolution law as shown in Figure 5.6 for the normal direction. In this study, a quadratic
nominal stress criterion was used to incorporate mixed mode conditions; this criterion is shown to
be successful in regards to prediction of delamination [132]. The criterion is defined as
2 2 2
〈𝑡 〉 𝑡 𝑡
𝑛 𝑠 𝑡
{ } +{ } +{ } = 1 (5.11)
𝑡0 𝑡0 𝑡0
𝑛 𝑠 𝑡
where the superscript 0 denotes the maximum traction or initiation traction value.
The Benzeggagh-Kenane (BK) [133] fracture energy criterion is used here with the assumption
that the critical fracture energy during separation along the first and the second shear direction are
the same i.e. 𝐺𝐶 = 𝐺𝐶.
𝑠 𝑡
𝜂
𝐺𝐶 +(𝐺𝐶 −𝐺𝐶){𝐺𝑆} = 𝐺𝐶 (5.12)
𝑛 𝑠 𝑛
𝐺𝑇
where 𝐺 =𝐺 +𝐺 and 𝐺𝐶 is the fracture energy in normal direction, 𝐺𝐶 is the fracture energy
𝑆 𝑠 𝑡 𝑛 𝑠
purely in the first shear direction (𝐺𝐶 and 𝐺𝐶 are assumed to be equal). The total fracture energy
𝑠 𝑡
in the mixed mode condition defines as 𝐺 = 𝐺 +𝐺 and 𝜂 is a cohesive property parameter [82,
𝑇 𝑛 𝑆
87, 134].
48 |
ADE | Determination of Cohesive Model Parameters
Carter and Evans [84] designed experimental setups to measure cement shear bond and the
hydraulic bond between casing and cement and demonstrated that the bond properties were both
pressure and temperature dependent. Shear bond is essential to support the pipe mechanically,
whereas the hydraulic bond prevents the formation of micro-annuli. Hydraulic bond failure may
happen due to casing expansion and contraction of a wellbore because of different wellbore
operations. They also designed another setup to measure the bonding properties between cement
and rock formation [82]. Evans and Carter [86] presented the push-out test which repeated by
Ladva, Craster, Jones, Goldsmith and Scott [85] using cement class G to measure the shear
bonding between cement and formation.
In the analyses performed in this contribution, we have adopted the cohesive parameters
determined by Wang and Taleghani [37]. They performed an inverse analysis on the experimental
results of [84-86] which are summarised in Table 5.2.
Table 5.2: Cohesive properties of cement / casing and cement/rock [37]
Shear
Cohesive Normal Cohesive Critical
Strength
Properties Interfaces Strength (kPa) Stiffness (kPa) energy (𝑱/𝒎𝟐)
(kPa)
Casing/Cement
2000 500 30𝐸6 100
Interface
Cement/Formation
420 420 30𝐸6 100
Interface
Whilst the parameters adopted have been determined from a number of experimental studies,
there remains great uncertainty regarding these parameters. Carter and Evans [84], Evans and
Carter [86] demonstrate that the cement shear bond to the casing is dependent on the curing
temperature, the pipe condition, and variations of different cement formulations. The adherence
degree of well cement to rock is highly variable and site dependent. The cement hydraulic bond
to the casing and formation is dependent on type of the formation, surface finish of the pipe, type
of mud layer, and degree of mud removal [85, 135].
5.4. Finite Element Modelling
A three-dimensional finite element framework is utilised to investigate the effect of pressure
increasing events such as pressure integrity testing on the cement sheath integrity using ABAQUS
49 |
ADE | / Standard software package [57]. Pressure tests are performed after the casing cementation, such
as casing integrity tests or formation integrity tests (leak-off test) by applying pressure upon
recently set cement [26]. In order to have more realistic simulations, stress-related factors which
induce wellbore failure in the fields especially during drilling operations were incorporated within
the framework including eccentricity and applying anisotropic in-situ stresses.
Material Properties
The cement sheath was modelled by using CDP model and calibrated according to experiments
performed on cement class G, as addressed in section 5.2.1. The interfaces of cement sheath with
casing and rock formation were modelled using surface-based cohesive behaviour feature using
cohesive parameters mentioned in Table 5.2. Elastic mechanical properties of the steel casing and
four different rock formations are defined as shown in Table 5.3. For ease of comparisons the
rocks’ stiffness were normalised with respect to the cement’s stiffness detailed in Table 5.3.
Table 5.3: Mechanical Properties of Casing and Rock Formation
Poisson’
Young’s
s ratio Casing
Casing Properties Modulus Reference
ν Grade
E (GPa)
Steel Casing 210 0.3 C-75 [37]
Poisson’ 𝑬
𝑵
Young’s
Formation Properties s ratio (Norma Reference
Modulus E (GPa)
ν lised)
Soft Rock 0.807 0.4 0.12 [136]
Shale 3.25 0.26 0.47 [37]
Hard Rock-1 17.2 0.2 2.51 [47]
Hard Rock-2 27.2 0.2 3.96 [47]
50 |
ADE | Geometry and Discretisation
The model consists of a casing, cement sheath with eccentricity, formation rock and the
interfaces of cement sheath with casing and formation shown in Figure 5.7. To reduce the
computational cost of the model, half symmetry has been exploited and a 5 in. horizontal slice
considered. The casing outside and inside diameter is chosen according to Schlumberger’s i-
Handbook [137] 7.625 in. and 6.625 in. respectively [47]. The borehole size is 8.5 in. The total
extent of the modelling of the surrounding formation is important to avoid any artificial effects in
the stress distributions and to assure that far-field stresses are applied from a reasonable distance
from the wellbore. According to Salehi [114], the model size should be at least four times bigger
Rock Formation
Casin
Cement
g
Sheath
Interface of
Interface of the Cement
Casing and
Sheath and the Rock
Cement
Figure 5.7: Casing, Cement Sheath With 70% Eccentricity, Formation Rock and the Cement Interfaces
with The Casing and Rock Formation
than the borehole size. Furthermore, the section near the wellbore has to mesh finer than the rest
of the formation. This finer section should be at least 2-3 times bigger than the borehole size to
improve accuracy [114]. Therefore, the formation rock was partitioned into two sections and
meshed with finer mesh near the wellbore area and coarser mesh in the far field area.
A mesh sensitivity analysis was performed to obtain a pragmatic element size in terms of
accuracy and computation time. The damage formulation in ABAQUS [57] alleviates the well-
known mesh dependency problem associated with local damage models by incorporating the
concept of characteristic length [56] into the formulation.
51 |
ADE | Initial State of Stress and Boundary Conditions
Initial geo-stress components were defined as 𝜎 and, 𝜎 in the initial step of the analyses.
𝐻 ℎ
Maximum and minimum horizontal stresses were applied parallel to X-axis and Y-axis in an
exchangeable way. The anisotropy of geo-stresses would cause shear stresses to the wellbore [82]
and is required to be considered in cement integrity modelling. The formation density and drilling
fluid density is assumed as 2000 kg/m3 and 1557.74 kg/m3 respectively [47]. The model thickness
(height of the model) in comparatively small to the width of the model, hence, the variation of
overburden (vertical) initial stress in depth is negligible and not considered in the model. The
corresponding overburden effective stress at the surface casing shoe with the vertical depth of 560
m was computed as 𝜎 =20 MPa, and all shear components are zero as shown in Figure 5.8.
𝑣
Displacement constrains were applied to the normal direction of bottom surface, outer surface of
formation, and symmetric surfaces.
Figure 5.8: Applying Anisotropic In-situ Stresses
5.5. Results and Discussions
A pressurized eccentric wellbore is subjected to isotropic and anisotropic in-situ stresses. Table
5.4 describes the three arrangements of applied in-situ stresses. The isotropic in-situ stresses were
applied as 𝜎 =𝜎 to have a basis results for comparison purposes and the anisotropic in-situ
𝐻 ℎ
stresses applied in Case-1 and Case-2.
The stress state, plastic deformations, and debonding within the cement sheath corresponding
to the different scenarios were analysed. The cement mechanical properties (given in Table 5.1),
the degree of eccentricity and the overburden pressure of 20 MPa are maintained constant for all
the scenarios. The contribution of surrounding rock formations' properties along with in-situ stress
52 |
ADE | confinement effects on cement mechanical failure was analysed by varying four different rock
formations' stiffness given in Table 5.3 (sections 5.5.1 and 5.5.2).
Table 5.4: In-situ Stress Arrangements
𝝈 𝝈 𝝈 = 𝝈 (MPa) 𝝈 = 𝝈 (MPa)
In-situ stress arrangements 𝑿𝑿 𝒀𝒀 𝑿𝑿 𝑯 𝒀𝒀 𝒉
Basis Case (Isotropic) 𝜎 𝜎 12.6 12.6
𝐻 𝐻
Case-1 (Anisotropic) 𝜎 𝜎 12.6 8.82
𝐻 ℎ
Case-2 (Anisotropic) 𝜎 𝜎 8.82 12.6
ℎ 𝐻
The effect of applied pressure along with different cases of in-situ stresses confinement and
different rock properties were analysed through interpretation of compression damage (crushing
index) and tensile damage (cracking index). The potential debonding occurrences were
investigated by using surface-based cohesive behaviour interaction property at the interfaces
without any pre-assumption of the crack initialisation or localisation propensity.
Compression Damage
The potential crushing caused by pressuring the wellbore along with in-situ stress within the
cement sheath is examined through the compression damage contours, local compression damage
paths and a global compression damage indicator in the following sections. Figure 5.9 illustrates
the compression damage contours within the cement sheath. The compression damage contour
within the cement sheath applying isotropic in-situ stresses (Basis-Case) for all the scenarios. with
different rocks’ properties shows the dominant effect of eccentricity on the stress distribution
within the cement sheath regardless of in-situ stress arrangements.
The effect of in-situ stress anisotropy is examined in Case-1 and Case-2 along with different
rock’s properties. As can be seen in Figure 5.9, the compression damage is mainly distributed at
the narrower side of the cement sheath. Due to different arrangements of in-situ stresses the
compression damage magnitude and the localisation of cracks bands changes for each case. This
effect is better visualised for scenarios case. This effect is better visualised for scenarios with
softer rocks E <1) in Case-1 and Case-2. As shown in the contour plots the magnitude and
N
propagation of compression damage for stiffer rocks (E >1) are similar for anisotropic cases.
N
53 |
ADE | Figure 5.11 shows compression damage along the cross-sectional paths for rocks’ stiffness
simulations at which 𝐸 <1 (Soft rock 𝐸 =0.12, Shale 𝐸 =0.47) considering three cases of
𝑁 𝑁 𝑁
applying in-situ stresses. As can be seen in all the scenarios considering two different rocks’
mechanical properties, the highest level of damage occurred within the path-1 (a) and path-1 (d)
located at the narrowest side of the cement sheath. The localisation and distribution of local
compression damage within the cement sheath for anisotropic cases surrounded by soft rock
Figure 5.10: Three Different Cross-Sectional Paths within the Cement Sheath
(E =0.12) and the shale formation (E =0.47) are similar. However, the magnitude of maximum
N N
compression damage is relatively higher for the softest rock and the local maximum compression
damage occurred within the first segment of the path-1 (a) which is the inner wall of the cement
sheath at one isolated node. Crushing failure potential in such a case (E =0.12) can be considered
N
higher than the case of shale formation (E =0.47). The shale formation is stiffer than the soft rock,
N
the resistance of the system becomes higher against pressure, therefore, the crushing damage index
is lower.
The magnitude of damage decreases as the path goes on towards the outer wall of the cement
sheath along the paths. Comparing graphs in Figure 5.11 shows the descending trend of the local
compression damage from the narrow side (path-1) towards the widest side (path-3).
55 |
ADE | (a) (b) (c)
(d) (e) (f)
Figure 5.11: Compression Damage along the Three Paths for
Simulations with 𝐸 <1 (vertical red lines indicate the corners)
𝑁
Figure 5.12 shows compression damage along the cross-sectional paths for rock’s stiffness
simulations at which E >1 (Hard Rock-1(E =2.51), Hard Rock-2 (E =3.96) considering three
N N N
cases of applying in-situ stresses.
Figures 5.12(a) and 5.12(d) demonstrate that the magnitude of maximum compression damage
is considerably lower for the Basis-Case (isotropic cases) in comparison with Case-1 and Case-2
(anisotropic cases). The considerable difference in maximum compression damage magnitude is
indicative of the destructive role of in-situ stress anisotropy on causing crushing damage within
the cement sheath.
The response of the cement sheath with stiffer rock and anisotropic in-situ stresses to the
elevated bore pressure for both rocks 𝐸 =2.51 and 𝐸 =3.96 was similar regarding local
𝑁 𝑁
compression damage magnitude and localisation. The magnitude of maximum compression
damage in situations in which rock is stiffer than cement (E > 1) in some nodes is quite
N
considerable (≈0.6) as it can be seen in Figure 5.12 (a) and (d). The local maximum compression
damage magnitude is still located within the narrower side but more distributed than the softer
56 |
ADE | Figure 5.13: Global Compression Damage Indicator vs. 𝐸
𝑁
Considering the contour plots, and the global compression damage indicator in Figure 5.13
confirms that compression damage was more distributed within the cement sheath in Case-1 and
Case-2 in comparison with the Basis-Case for scenarios involving the softer rocks (E <1). While
N
the global compression damage indicator for scenarios with stiffer rocks (E >1) was similar in all
N
the cases regardless of in-situ stress arrangements. The different response of the systems with
stiffer rocks (E >1) towards in-situ stress arrangements indicate that although the anisotropy of
N
in-situ stresses in Case-1 and Case-2 imposes the additional shear stress to the system, the stiffer
rocks possess higher resistance against the shear stress and don’t transfer these stresses to the
cement sheath.
Tensile Damage
The tensile cracking susceptibility is examined through the tensile damage contours and a global
tensile damage indicator. The state of local tensile damage (cracking) contours within the cement
sheath after applying isotropic and anisotropic in-situ stresses is shown in Figure 5.14. As can be
seen, the tensile damage is more localised in comparison with compression damage shown in
Figure 5.9. Considering the tensile damage contours of the Basis-Case demonstrates the important
role of eccentricity in the distribution of tensile stress within the cement sheath again.
58 |
ADE | Figure 5.14: Tensile Damage Contours within the Cement Sheath
The maximum tensile damage for the Basis-Case reached 0.48 for the softest rock (E =0.12).
N
In contrast, the maximum tensile damage magnitude reached 0.67 and 0.58 for the softest rock in
Case-1 and Case-2 respectively and covered a relatively large zone on the narrowest side. The
high difference of tensile damage magnitude shows the critical effect of in-situ stresses anisotropy
particularly in scenarios with softer rocks (E <1). Tensile damage contours in Figure 5.14 show
N
for stiffer rocks scenarios (E >1), the tensile damage magnitude and its localisation are similar
N
for all the scenarios.
The surface-based cohesive behaviour defined by means of the contact interaction property
enables the interface of cement sheath to transmit normal and shear forces across the interface as
described in Equation 10. The tangential slips of the interface are assumed elastic, and it is resisted
by the cohesive strength of the bond while the cohesive stiffness is undamaged which leads to the
creation of shear forces. The degradation of cohesive stiffness and evolution of damage in shear
directions defined in Equations 5.11 and 5.12 as well. The simulations show high contact shear
stresses at the interface of the cement and the rock formation is the driving force for initialisation
59 |
ADE | and propagation of tensile cracks through the whole thickness of the narrow side of the cement
sheath.
To compare all the zones within the cement sheath experiencing tensile cracking, a global
tensile damage indicator (D ) was computed as follows.
t
𝐷
=∑𝑁(𝑑𝑡)
(5.14)
𝑡 0
𝑁
where d is the local tensile damage magnitude for all the nodes within the cement sheath, and N
t
is the number of nodes with associated tensile damage.
Figure 5.15: Global Tensile Damage Indicator vs. 𝐸
𝑁
Figure 5.15 shows tha t the cement sheath surrounded by the softest rock (E =0.12) experiences
N
the highest level of tensile damage the in anisotropic cases (Case-1 and Case-2). The low stiffness
of the rock makes the cement sheath more vulnerable to the additional shear stresses caused by
anisotropic in-situ stresses and results in the formation of microcracking. While in situations with
stiffer rocks (E >1), the lower level of microcracking is seen in Figure 5.15 due to the high
N
resistance of stiffer rocks against the shear stresses.
The simulations confirm the significance of tensile cracks and tensile properties to be
incorporated into the constitutive modelling. In situations in which E <1(softer rocks) as shown
N
in Figure 5.14, the relatively high magnitude of tensile damage (above 0.5) means almost above
50% of tensile strength was degraded and significant tensile cracks initiated and propagated
through the whole thickness of the cement sheath.
60 |
ADE | Propensity of Forming Micro Annuli
The soundness of the cement sheath bonds with the casing and the rock formation is examined
through a contact stiffness degradation index. Figure 5.16 demonstrates the starting location of the
selected paths along the cement sheath interfaces with the casing and rock formations.
Figure 5.16: Cement Sheath Interfaces with the Casing and Rock Formation
The contact stiffness of the cement sheath with the casing is fully degraded for all the
combinations of rock properties shown in Figure 5.17 regardless of in-situ stress arrangements. In
contrast, the degradation of cement sheath bond with the rock formation is dependent contrast, the
degradation of cement sheath bond with the rock formation is dependent on the rock‘s stiffness
shown in Figure 5.18.
Figure 5.17: Contact Stiffness Degradation at Cement Sheath Interface with the Casing
T he response of the cement sheath interface with rock formation for the softest rock (E N=0.12)
is different than the rest of the scenarios (Figure 5.18). In Case-1 the due to high confinement of
in-situ stresses the narrower part show more resistance to sliding as can be seen at the beginning
of the interface length while in Case-2 the effect of high confinement can be seen in the middle
length of the interface. In the Basis-Case (Figure 5.18(b)) the contact stiffness is undamaged for
some sections at the widest section of the cement sheath.
61 |
ADE | (a) (b)
(c)
Figure 5.18: Contact Stiffness Degradation at Cement Sheath Interface with the Rock Formation
The interfaces are the most vulnerable part of a wellbore due to the high difference in the
stiffness of surrounding materials, and high contact shear stresses in tangential and normal
directions of the interface length in wellbore operations. The contact stiffness degradation,
reaching near to one in all the simulations, indicates the high potential to debond. Cement sheath
centralisation, remedial cementing and using expandable liners (at the interfaces of cement and
casing) may mitigate these effects.
5.6. Conclusion
A systematic approach was taken to assess the integrity of cement sheath after being pressurised
in relation to the creation of cracks within the cement sheath and microannulus made of class G
well cement. The key point of the approach is the employment of a constitutive model taking into
account the difference in tensile and compressive responses and the pressure-dependent of the
behaviour under shearing at different levels of confinement. To obtain the corresponding model
parameters, laboratory experiments, including uniaxial compression tests and three-point bending
tests, were performed on specimens manufactured from class G well cement.
A three-dimensional finite element model including a casing, cement sheath with eccentricity,
and rock formation was built to investigate the effect of enhancing pressure in a wellbore. The
interfaces of cement sheath with the casing and the rock formation were modelled using surface-
based cohesive behaviour to examine debonding. The integrity of the cement sheath and the
interfaces were investigated through different scenarios of changing in-situ stress orientations and
different rocks’ stiffness.
The results show the dominant effect of eccentricity on the distribution of stress within the
cement sheath which emphasises the importance of casing centralisation. Comparing the damaged
area and geometry of cracks in anisotropic in-situ stresses scenarios with isotropic scenarios
suggests that wellbore operations require more attention within the heterogenic geological fields.
62 |
ADE | Effect of Curing Conditions on the Mechanical Properties of Cement
Class G with the Application to Wellbore Integrity
ABSTRACT
Wellbore integrity is highly dependent on the cement sheath integrity. Cement sheaths play an
essential role in preventing any communication between the formation fluids and the surrounding
environment. Mechanical failure of the cement sheath within a wellbore is influenced and
governed by many factors including cement mechanical properties. However, the paucity of
cement class G mechanical parameters including lack of experimental data under different
confining pressure, tensile properties, and the effect of curing temperatures on the long-term
cement mechanical properties are impediments to the numerical simulations in wellbore integrity
assessments. Therefore, this study aims to expand the cement class G mechanical properties
inventory. This paper investigates the mechanical behaviour of cement class G at two different
curing temperatures (30oC and 70oC) at the age of 28 days. The effect of both the curing regime
and confining pressures (15 MPa and 30 MPa) on the strength and post-peak response of the
cement under compression are examined. The measurement of tensile capacity and fracture energy
performing indirect three-point bending tests along with the challenges involved with measuring
fracture energy and modifications incorporated to the three-point bending test set-up, to measure
fracture energy properly, are explored. The results were validated by Digital Image Correlation
(DIC) technique measurements. The obtained experimental were interpreted and subsequently
utilised as input data for a constitutive model specifically formulated for modelling the geo-
materials such as cementitious materials and validated by numerical analysis.
6.1. Introduction
Wellbores provide access to natural resources such as oil and gas and are encased in concentric
layers of steel casing and cement sheaths. After drilling the borehole, steel casing is inserted and
is held in place and protected by a sheath of cement which is pumped into the annular gaps.
Although wellbores are sealed and block any interaction which may occur between formation fluid
and geologic strata, the integrity of wellbores might still be compromised [9, 18]. At this stage,
wellbores may turn into the high-permeability conduits for the formation fluids [19] which could
pose a potential risk to the environment by contaminating groundwater and/or the atmosphere.
The cement sheath is responsible for providing zonal isolation and preventing the leakage of
formation fluids during the lifetime of a wellbore [31] and therefore the cement sheath should be
67 |
ADE | designed and placed so that it withstands the external conditions imposed upon it, including, in-
situ stresses, high internal pressures and high temperature.
Portland Cement Class G is mostly utilised as the base of oil wells in the oil and gas industry.
Additives may also be incorporated to achieve certain properties [34]. The general components of
cement class G are general grinding Portland cement clinker, Dicalcium Silicate (Ca SiO ) and
2 4
water. During the manufacture of cement class G, only Calcium Sulfate (CaSO ) and water can
4
be added to the cement clinker [42]. The mechanical properties of the cement are highly dependent
on the curing conditions, which vary along the wellbore depth and corresponding to the exposure
to the formation fluids with different conditions [20, 88, 89]. It is worth noting that the cement
used in the oil and gas industry has very low permeability, usually less than a 0.2 mD [38].
Therefore, hydraulic isolation is attained straightforwardly, and any probable leakage can only
happen through mechanical failures of the cement sheath [11].
The behaviour of the cement sheath under different conditions should be properly simulated
using an appropriate constitutive model to predict the mechanical damage of the cement sheath.
The effect of pressure and temperature changes, which may result in shear/compression
(crushing), and tensile (cracking) damage should be incorporated into the constitutive model. One
of the most challenging parts of constitutive modelling is to determine the model parameters
through performing experiments and interpretation of the experimental outcomes. Many
laboratory tests have been carried out on well cement, simulating the wellbore condition, to
determine the key parameters for modelling purposes.
Thiercelin, Dargaud, Baret and Rodriguez [108] performed a study on cement class G, with
varieties of additives, to determine the material's flexural, compression strength, and Young's
modulus in flexural and compression. The tensile properties were obtained using three-point
bending tests on 30×30×120 mm prisms, with a loading rate of 0.01 cm/min. The compressive
properties were measured via uniaxial compression tests on 50.8×50.8×50.8 mm (2×2×2 in) cubes.
The volume of additives, curing conditions, and slurry density were different for each test.
Therefore, it is hard to associate any change in mechanical properties with one specific factor.
Roy-Delage, Baumgarte, Thiercelin and Vidick [91] designed a slurry formulation with cement
class G to achieve highly durable cement. They cured the samples at 77oC and 114oC with a
pressure of 20.7 MPa (3000 psi) for three days or upon reaching a constant compressive strength.
Three-point bending tests and crushing tests were performed on beams with the size of 30×30×120
mm and cubes with the size of 50.8× 50.8× 50.8 mm (2× 2× 2 in) respectively. They investigated
68 |
ADE | the interaction between flexible cement and expanding agents and concluded that the cement with
both flexible and expanding additives shows more durability in long-term periods.
Cyclic pressure tests were run on hollow cylinders (50×100 mm) of cement class G by Yuan,
Teodoriu and Schubert [103]. The samples were cured under three different conditions for 14 days:
room conditions; atmospheric pressure 0.1 MPa (14.7 psi) at the temperature of 75oC in an oven;
and under 18 MPa (2610 psi) at 100oC in an autoclave. They demonstrated the cement can endure
more cycles as confining pressure increases. Additionally, the cement with a higher Poisson's ratio
and lower Young's modulus is stronger encountering low cycle fatigue.
Nasvi, Ranjith and Sanjayan [92] used cylindrical samples of 50 ×100 mm to measure the
uniaxial compression strength of cement class G. The samples were oven cured at different
temperatures between 300C to 800C for 24 hours excluding the samples required to be cured at
room temperature. Subsequently, all of the samples were kept at ambient temperature for another
48 hours. Their results demonstrated that samples that cured at 600 C had the maximum uniaxial
compressive strength of 53 MPa, but that samples cured above this temperature yielded a lower
uniaxial compressive strength. The Young's modulus of cement class G is higher at lower curing
temperatures and reaches its maximum value at the curing temperature of 400 C.
Guner and Ozturk [94] measured both uniaxial compressive strength and Young's modulus at
different cement curing periods of 2, 7, and 14 days. They concluded increasing the curing time
increases the mechanical properties of cement by 2-3 times.
Teodoriu and Asamba [95] investigated the effect of salt concentration on cement class G
properties by performing uniaxial compression tests on cubic samples with the size of 50.8× 50.8×
50.8 mm (2× 2× 2 in). They cured samples in water in atmospheric condition for 24hrs and then
the samples were placed in an autoclave for curing period of one to seven days under two different
conditions (30oC and 10 MPa / 150oC and 20 MPa). They also measured the compressive strength
of samples cured at the atmospheric condition at the age of 21 days. Their results of the batch
without salt with respect to the first curing condition were summarised in Table 1 for comparison
purposes. They showed the samples with 5% ± 2.5% salt concentration curing at ambient to
moderate temperature, yield the maximum compressive strength among all the other samples with
different salt concentration curing at different conditions.
Romanowski, Ichim and Teodoriu [96] compared two methods for measuring the cement
compressive strength (ultrasonic pulse velocity versus mechanical method). The tests were
performed at different curing times on cement class G, cement class G with bentonite, and cement
class G with other additives. They demonstrated that the outcomes of ultrasonic methods should
69 |
ADE | Rogers [99] continued their study by curing samples at two different temperatures of 54.40C or
82.20C for 48 hours in an atmospheric water bath. The authors compared the results of splitting
tensile strength (STS) tests with direct tensile tests on the dog-bone sample. The splitting tensile
strength test results overestimated the tensile properties of cement class G by order of 1.5 to 2.5.
However, direct tensile test measurements can also be impacted by stress concentrations on the
samples at or near grip points (point loading), which can lead to immature breakable of the
samples.
Dillenbeck, Boncan, Clemente and Rogers [98] performed uniaxial tensile tests on dog-bone
samples made of cement class H and additives to measure the cement uniaxial tensile strength.
They developed a new testing machine to simulate downhole conditions in a wellbore for curing
purposes and performed tensile tests on dog-bone samples. The results showed that the uniaxial
tensile strength of the cement samples was highly dependent on the stress loading rate. Therefore,
the authors addressed the necessity of developing a standard loading rate at which to perform
cement tensile tests.
Quercia, Chan and Luke [100] characterised the tensile strength of cement class G using the
Weibull method by performing direct tensile tests on dog-bone samples, and Brazilian tests on
cylindrical samples. Weibull statistics is a characterization tool which describes the strengths
spread along with strength variations due to sample size and provides more assurance and
reliability in risk analysis [100]. The samples were made of cement class G and micro-fibres. The
samples were initially cured in a pressurised chamber for 24 hours at 250C and 10.34 MPa (1500
psi) and then demolded and cured underwater for six days. The authors also used modified dog-
bone molds, which act as holders to avoid grip concentration points. Table 6.2 summarises the
results of tensile test studies that are available in the literature to the best of our knowledge.
74 |
ADE | values (the uniaxial strength measured using sufficiently slender specimens is usually around
70%-90% of the cube strength [106]. In cubic samples, the restraining effect of the platens spreads
over the total height of a specimen, but in cylindrical samples, some parts of specimens stay
unaffected [107]. Another problem regarding using cubes is that the post-peak behaviour is milder
in cubic specimens, therefore, requires more energy consumption than using cylinderial
specimens. Additionally, the effect of specimen size is also larger for cubic samples [105].
Table 6.2 indicates that the measurement of tensile strength and particularly the fracture energy
of cement class G, in particular over long-term periods, were simply overlooked in many
experimental studies. The main problem with performing tensile measurement test is there are no
API guidelines for measuring the tensile properties of cement, and ASTM standards for the
measurement of tensile properties present various limitations when applied to cement tensile tests.
This is because, these standards have been designed for cement cured at locations only a few
meters down the ground and they do not incorporate the curing conditions considering downhole
conditions in regards to high pressure and high temperature in harsh conditions, i.e. downhole
conditions [97, 98]. This paucity of a complete inventory of cement class G mechanical properties
can be an obstacle to performing precise integrity simulations. Subsequently, in this study, three-
point bending tests were performed on notched and un-notched beams to measure the tensile
strength and fracture energy.
The selection of an appropriate constitutive model for the cement as a geo-material and its
corresponding failure surface parameters are the utmost of importance part of wellbore integrity
modelling. This paper presents the results of an experimental study designed to fill the gaps above
relating to the responses of cement class G in unconfined, confined compression tests, three-point
bending tests. The results of the tests were interpreted to obtain the failure envelope of a
constitutive model (Concrete Damage Plasticity Model [54, 115]) that was particularly formulated
for quasi-brittle behaviour modelling. The model considers the differences in tensile and
compression responses and the pressure-dependent nature of the cement’s behaviour under
shearing at different levels of confinement. In order to simulate the tensile response of the cement
class G and address the gaps regarding fracture energy measurements, three-point bending tests
were carried out using a modified method to obtain fracture energy by performing three-point
bending tests.
This paper is organised as follows; section 6.2 describes the used material, the procedures
curing and samples preparation. The effect of curing temperature on the cement mechanical
properties is investigated in section 6.3 by performing the unconfined and confined tests on
76 |
ADE | samples cured at two different curing temperatures (30oC and 70oC). Section 3 also describes the
execution of the three-point bending test on the prismatic samples cured at 30oC and the
challenges involved with measuring fracture energy. Modifications are incorporated to the three-
point bending test set-up, explained in section 6.4, to facilitate the measurement of fracture energy,
followed by validation of the test performance by using Digital Image Correlations (DIC) cameras.
In section 6.5 the outcomes of compression tests were interpreted to obtain the shape of yield
surface of the Concrete Damage Plasticity (CDP) Model, followed by the curve fitting procedures
to obtain the corresponding parameters and validation of them by the numerical analysis. Section
6.6 concludes the paper thereafter.
6.2. Material and Sample Preparation
The chemical and physical properties of the cement class G used in this study can be seen in
Table 6.3. The slurry density was 1900 kg/m3, corresponding to a water-to-cement mass ratio of
0.44. The slurry was prepared according to API-10 [138]. The cement class G samples were cured
in a pre-heated water bath at two different curing temperatures (30oC and 70oC), for 28 days, to
examine the effects of curing temperature on the mechanical properties of the cement class G.
Table 6.3: Cement Class G Components
Typical
Chemical Properties
values %
Sulfuric Anhydride (SO3) 2.7
Magnesia (MgO) 1.1
Tricalcium Aluminate (C3A)+ 15.5
Tetra Calcium Alumino Ferrite
(C4AF)
Tricalcium Aluminate (C3A) 1.2
Tricalcium Silicate (C3S) 60
Physical Properties
Specific gravity 3.18
Free Fluid Content (%) 4.5
Sample preparation
Samples were prepared in both cylindrical and prismatic shapes. The cylinder samples, at 42
mm diameter and a length of 100 mm, were used in compression tests. The prismatic samples, at
77 |
ADE | investigated by testing two slower rates: 0.1 mm/min and 0.04 mm/min, shown in Figure 6.1. The
axial displacement of the loading platen was measured with the help of two external, linear,
variable differential transformers (LVDT) that were installed 180oapart at the top platen.
As can be seen in Figure 6.1, the samples showed very brittle behaviour at a displacement rate
of 0.2 mm/min and 0.1 mm/min. To capture post-peak behaviour, the displacement rate was
reduced to 0.04 mm/min, at which rate the specimens showed less brittle behaviour. The slowest
rate was therefore chosen as our settled displacement rate.
Confined Compression Tests at the Curing Temperature of 30°C
Triaxial compression tests with confining pressures (P) of 15 MPa and 30 MPa were performed
c
after 28 days, on the samples that were cured at 300C. The loading path was designed so that
samples reached the desired confining pressure at the first step and were then loaded axially under
displacement control until failure occurred. Figure 6.2 compares the results of two different
confining pressures for specimens cured at 300C.
Figure 6.2: Axial Stress-Strain Response for Two Different Confining Pressures at Curing Temperature of
30°C
Figure 6.2 shows that the load-carrying capacity of the cement under higher confining pressure
is significantly higher for the same axial strain. The response of the cement in the triaxial test does
not illustrate a well-defined peak; the slope of the graph gradually decreased until it almost reached
a plateau. The hardening process decreasing with an increased axial load is indicative of an
increase in the effective compressive strength and ductility with confinement. This ductile
behaviour at larger strains creates two macro-cracks without other distributed micro-cracks.
79 |
ADE | Unconfined Compression Test at a Curing Temperature of 70°C
To investigate the effects of curing temperatures on the mechanical properties of the cement,
the samples were prepared as described in section 6.2.1 except for a change in curing temperature.
For these tests, the temperature was set to 70oC. The rest of the test conditions, including the size
of the samples, the curing period, and the water-to-cement ratio, were kept constant. Figure 6.3
shows the stress-strain curve obtained for three uniaxial compression tests on specimens cured
at 70oC over 28 days.
Figure 6.3: Axial Stress-Strain Response under Uniaxial Compression Test at Curing Temperature of 70°C
Comparing Figures 6.1 and 6.3 shows that increasing the curing temperature to 70oC, leads to
an almost 27% strength reduction in uniaxial compression strength compared with curing at 30oC.
The strength reduction at higher temperatures can be attributed to an increase in early strength,
but a decrease in later strength (microstructural effect)[139]. Calcium Silicate Hydrates (C-S-H)
is denser at higher temperatures, and thus occupies less volume, which leads to more porosity and
less strength. Correspondingly, above 50°C ettringite becomes unstable and occupies less volume
as well, which again leads to strength reduction [139].
Confined Compression Tests at Curing Temperatures of 70°C
Figure 6.4 shows the results obtained from triaxial tests with a confining pressure of 15 MPa
on specimens cured at 70oC. The sudden termination of the plateau might be related to the quick
propagation of macro-cracks, or to the breakage of the liner (membrane) within the Hoek cell,
which leads to the entrance of oil and a loss of confinement.
80 |
ADE | and the graph obtained from the three-point bending test on one of the un-notched beams, are
shown in Figure 6.5.
The tensile strength σ for prisms was calculated from the bending tests, as follows.
t0
3𝐹𝑆
𝜎 = (6.1)
𝑡0 2𝑑 𝑑2
1 2
where F is the maximum load, S is the span of the beam, d is the width and d is the depth of the
1 2
beam [141]. As can be seen in Figure 6.5, the beam samples showed a highly brittle behaviour.
A notch of 5 ×15 mm was cut at the centre of the beams to measure the fracture energy
according to the RILEM recommendations [140], on “Determination of the fracture energy of
mortar and concrete by means of three-point bend tests on notched beams”. The notch was cut 15
mm to meet the requirement of the RILEM standard, which states that the notch depth should be
equal to half of the beam depth ± 5 mm. To record the load versus crack-mouth opening, a clip
gauge (crack extensometer) was installed using two plastic plates glued to the bottom surface of
the beam on each side of the notch. The rest of the test conditions, including the arrangement of
LVDTs and the displacement rate, were kept identical to performing three-point bending tests on
un-notched beams. The geometry of the notched beams, and the graph obtained from the three-
point bending test on one of the notched beams is shown in Figure 6.6.
D (Displacement rate of 0.015mm/min)
40mm
M 40mm
120 mm
Figure 6.6: Measuring Fracture Energy of Cement Class G using Notched Beams
Fracture energy is calculated according to RILEM TC 50-FMC [140] using the below
formulation.
𝑈 +𝑚𝑔𝑑
𝑜 𝑜
𝐺 = (6.2)
𝑓 𝐴
82 |
ADE | where U is the area under the load-deflection is graph; A is the ligament area and defines as A =
o
B(W−a ); B is the width of the beam; W is the depth of the beam; a is the initial depth of the
o o
notch; mg is the weight of the beam; and d is the final deflection at the load point.
o
As can be seen in Figure 6.6, the sample showed highly brittle behaviour, and the area under
the load-displacement graph is very small.
The displacement rate was the slowest rate that the MTS machine could function. As the pattern
repeated for all of the samples, the present authors concluded that there are some modifications
required to be able to measure the fracture energy of cement class G.
6.4. Modification of the Three-Point Bending Test Configurations to Measure
Fracture Energy
A different approach was undertaken to modify the test configuration so that we could capture
the post-peak response. The testing device was a three-point bending set-up, mounted on a servo-
hydraulic testing machine. The axial displacement of the loading platen was measured using two
LVDTs installed on both sides of the beam specimens. To prevent post-peak, highly brittle crack
propagation, the displacement rate was controlled by opening the crack mouth clip gauge instead
of the crosshead displacement. The set-up configuration is shown in Figure 6.7.
Load Set Point= 0.1 kN
240 mm
75 mm
75 mm
MTS
Clip Gauge Displacement Opening Rate = 0.001mm/min
Figure 6.7: Modification on Performing Three-point Bending Test
The clip gauge was installed between two plastic plates glued to the bottom surface of the beam,
on each side of the notch. Until the applied load reached 0.1 kN the displacement rate was
controlled by the crosshead, then the displacement rate was transferred to the opening of the crack
mouth by the clip gauge. A non-contact strain measurement technique using two-dimensional
digital image correlation (DIC) was applied simultaneously. The details of how we applied this
optical technique are described in section 6.4.3.
83 |
ADE | For a beam in a three-point bending test, the load-deflection graph consists of three stages. In
the first stage, the deflection rises linearly, as the load increases. A fracture process zone develops
during the second stage, at which micro-cracks are created. In the third stage, which is a strain-
softening zone, cracks grow quickly [33]. The fracture energy is calculated according to Equation
(6.2).
Sample Preparation
The beam samples with the size of 280×75×75 mm were used. The specimens were cast in steel
moulds in the laboratory. The slurry density and water-to-cement mass ratio were kept the same
as the previous experiments. After casting, the samples were covered with a wet burlap for 24
hours. On the second day, all the samples were de-molded and transferred to a fog room. The
samples were taken out of the fog room four hours prior to testing, and a saw-cut notch of 5 ×30
mm was made at the centre of the beams span.
Results
The obtained graph from one of the experiments is shown in the graph in Figure 6.8. The
computed fracture energy, after three repeats of the test were performed, were 21, 24.64, and 23.44
N/m.
Digital Image Correlation (DIC)
Digital image correlation (DIC) is an optical and non-contact surface-displacement
measurement technique. In this technique, surface images before and during the deformation are
taken by digital cameras. Operating this technique allows the computation of any point
displacements on the sample by the corresponding computer software using the taken images
84
Figure 6.8: Measuring Fracture Energy of Cement Class G Undertaking the
Modified Approach |
ADE | before and during loading process [142-144]. The DIC system (3D) used in the experiments
consisted of two monochrome 2.8-megapixel, conventional charge-coupled device (CCD)
cameras. It had a sensor size of 1/1.8" and a maximum resolution of 1928×1448 pixels. The camera
lens was a 75-mm Fujifilm prime lens with an aperture size range of 1/22-1/2.8. This lens has a
minimal distortion, therefore, no correction for distortion was necessary. The camera body had a
Universal Serial Bus (USB) 3 interface for the fast and reliable image transfer. The cameras were
connected to a computer utilizing two software (Vic-Snap and Vic-2D) produced by Correlated
Solutions. The Vic-Snap software arranges the process of capturing images while the specimens
undergoes deformation. The Vic-snap software is utilised during calibration process and data
analysing.
The deformation measurements were based on the displacements of random speckles spread
over the surface of the sample. The speckle patterns should be applied in such a way to create
contrast, by painting the whole surface of the sample with white, and then creating black speckles.
The speckle pattern should be non-repetitive, well-distributed, and high contrast, to avoid any bias
measurement, or sensitive defocus [142, 143, 145]. Figure 6.9 shows a typical speckle pattern on
one the sample.
Figure 6.9: Speckle Pattern on a Prismatic Sample
To analyse the images after the test, an area of interest was chosen in which to detect the
deformations and strain localizations. The surface displacement was computed by comparing the
number of digital images taken during the test with the reference image (undeformed image). The
correlation computations were based on tracing the same pixel points placed in different images.
The displacement field inside a pattern is presumed to be homogenous [144]. The initial reference
image, which is indicative of the undeformed body, is interpreted as a discrete function of p(x ,y);
i j
it is converted into another discrete function of p′(x′,y′) after deformation. In order to compute
the displacement of point p, a reference square subset of (2M+1)× (2M+1) pixels containing point
85 |
ADE | p (x ,y ) from the reference image is selected and used for tracking the associated displacement
o o
in the deformed image [143]. The relationship between these two functions is defined as follows
[144] .
𝑝′(𝑥′,𝑦′)−𝑝(𝑥+𝑢(𝑥,𝑦),𝑦+𝑣(𝑥,𝑦))= 0 (6.3)
where u(x,y) and v(x,y) are the displacement field for a pattern as shown in Figure 6.10. The
p′(x′,y′) coordination which is corresponding to the coordination of point o(x ,y ) shown in
o o
Figure 6.10 in the reference image can be computed as follows.
𝜕𝑢 𝜕𝑢
𝑥′ = 𝑥 +𝛥𝑥+𝑢+ 𝛥𝑥+ 𝛥𝑦 (6.4)
𝑜 𝜕𝑥 𝜕𝑦
𝜕𝑣 𝜕𝑣
𝑦′ = 𝑦 +𝛥𝑦+𝑣+ 𝛥𝑥+ 𝛥𝑦 (6.5)
𝑜 𝜕𝑥 𝜕𝑦
∂u ∂u ∂v ∂v
where u, v are the displacements of the subset centre point o in x, y-direction, , , ,and
∂x ∂y ∂x ∂y
are displacement gradients for the subset as shown in Figure 6.10.
𝑦 𝑦′
𝑦′
0
Deformed Sub- 𝑝′ (𝑥′, 𝑦′)
Reference Sub-image
image
v
Δ𝑦 𝑝 (𝑥,𝑦)
𝑦 0 𝑦 0
𝑥 𝑥′
𝑥 0 Δ𝑥 𝑥 0 u 𝑥 0′
Reference Image Deformed Image
Figure 6.10: Reference image and deformed image schematics after [142]
A two-dimensional DIC technique was chosen for this study. To have an accurate 2D-image
correlation, the alignment of the camera and specimen is crucial. The camera was set up planar
and parallel to the specimen. The images were captured by Vic-snap software using an exposure
time of 100 ms. Prior to the three-point bending test, the undeformed reference picture was taken,
86 |
ADE | along with a 20 mm pitch grid for calibration purposes. To examine the degrees of similarity
between the reference and deformed image, a correlation criterion should be employed [143]. The
default criterion is Normalised Sum of Square Difference (NSSD) [143] defines as follows.
𝑀 𝑀 𝑝(𝑥 ,𝑦 ) 𝑝′(𝑥 ′,𝑦 ′) 2
𝑖 𝑗 𝑖 𝑖
𝐶 = ∑ ∑ [ − ] (𝑖,𝑗 = −𝑀:𝑀) (6.6)
𝑁𝑆𝑆𝐷 𝑝̅ 𝑝̅′
𝑖=−𝑀𝑗=−𝑀
where
𝑝̅ = √∑𝑀 ∑𝑀 [𝑝(𝑥 ,𝑦 )]2 and 𝑝̅′ = √∑𝑀 ∑𝑀 [ 𝑝′(𝑥 ′,𝑦 ′)]2 (6.7)
𝑖=−𝑀 𝑗=−𝑀 𝑖 𝑗 𝑖=−𝑀 𝑗=−𝑀 𝑖 𝑖
The advantage of this DIC technique is that the displacement of crack openings can be measured
in different directions on the surface. In this study, to validate the results obtained from the clip
gauge (crack extensometer), the crack mouth opening displacements (CMOD) were also computed
using a DIC technique (optical extensometer), as shown in Figure 6.11.
Figure 6.11: Optical Extensometer at the Crack Mouth using DIC Inspector Tool
Figure 6.12 demonstrates the results obtained from the clip gauge and DIC are in good
agreement. However, the results from DIC show lesser displacement comparing to the clip gauge
which seems possible. The clip gauge was installed at the centre of the beam along the beam
thickness while DIC measurements are computed based on surface speckles displacements.
87 |
ADE | 0.8 0.2
Load
Clip Gauge
DIC
0.6 0.15
)
) m
N m
k ( d a0.4 0.1 ( D O
o M
L
C
0.2 0.05
0
0 5 10 15 20
250
Image Number
Figure 6.12: Load and CMOD versus DIC during the Three-point Bending Test
6.5. Interpretation of Results in the Concrete Damage Plasticity Model Framework
The non-linear behaviour of cement under compression can be modelled by plasticity or damage
approaches, or a combination of both. Plasticity is described by means of the unrecoverable
deformation after removing the load; damage is defined by the elastic stiffness reduction. Cement
under compression exhibits both plasticity and damage [4]. Therefore, in order to simulate
cement’s mechanical behaviour under compression, it is of the utmost of importance to use a
model in which both the plasticity and damage concepts have been embedded.
The stress-strain relationship subjected to uniaxial monotonic compression can be defined as:
𝜎 = (1−𝑑)𝐸 (ɛ−ɛ ) (6.8)
0 𝑝𝑙
where d shows damage variable; E is initial undamaged stiffness and is ɛ compressive
0 pl
equivalent plastic strains.
Yield Criterion
The yield function proposed by Lubliner, Oliver, Oller and Onate [54] and modified by Lee and
Fenves [115] defines in I and √J plane as:
1 2
1
𝐹(𝜎) = [𝛼𝐼 +√3𝐽 +𝛽〈𝜎 〉−𝛾〈𝜎 〉]= 𝑐(𝜅 ) (6.9)
1 2 𝑚𝑎𝑥 𝑚𝑎𝑥 𝑐
1−𝛼
where α, β and γ are dimensionless constants calibrated by experiments; c is the compressive
cohesion and its evolution is determined by uniaxial compression tests; κ is the hardening-
c
damage parameter; The cohesion stress c is to be scaled as its initial value is equal to initial yield
strength in uniaxial compression (f ). Subsequently, c = f when κ = 0 and c = 0 when κ =
co co c c
88 |
ADE | The parameters on the yield surface can be obtained using the experimental results. To
approximate the shape of the loading/yield surface in I and √3J plane, the results of compression
1 2
tests shown in Figure 6.15 (on samples cured at 30𝑜C) at different confinements were used.
Figure 6.15: Compression Tests on Cylindrical Samples Cured at 30°C
The points in Figure 6.16 represent the initial yield strength (the turning point for each
compression test in Figure 6.15). The points were fitted by the solid line in the graph to create the
f
approximate shape of yield surface which delivers α = 0.08 and γ = 0.683. Accordingly, bo can
fc
f
be computed using Equation (6.10) which results in bo=1.10 and β can be computed based on
fc
Equation (6.11) which yields to 27.8, and K can be computed as 0.84 using Equation (6.12).
c
Dilation angle ψ can be computed using the slope of the line in the I -√J plane as shown in
1 2
Figure 6.14 using triaxial compression tests resulted ψ =48.62o.
91 |
ADE | deform during the simulations. The defined boundary condition for the bottom surface constrains
all the degrees of freedom, and the top surface displacement rate was applied in the direction of
the cylinder axis.
The results and corresponding failure patterns in the laboratory and ABAQUS are shown in
Figures 6.17 and 6.18. The displacement rate was applied in the normal direction of the upper
surface. As can be seen, the simulation results can match the experimental counterparts in terms
of both macro responses and failure pattern, demonstrating that the obtained set of parameters are
appropriate for the modelling of the cement class G.
6.6. Conclusion
The cement sheath in wellbores is responsible for providing complete zonal isolation and
maintaining the integrity of the wellbores. However, the cement class G inventory lacks some
important aspects of cement mechanical properties which add uncertainty in the execution of
numerical simulations of wellbore integrity assessments. This study intends to expand the
inventory by performing compression tests considering different curing temperatures and indirect
tensile tests.
The experimental results for confined and unconfined compression tests on cylindrical
specimens of 42×100 mm, cured over 28 days at different temperatures (30oC and 70oC) were
performed in this study. The key findings of the uniaxial compression tests were:
• A suitable loading rate was determined for uniaxial tests in order to achieve a converged
post-peak response. The peak load was found to be dependent on the loading rate.
• Under uniaxial compression (unconfined), the response of the samples was
accompanied with a well-defined peak load, followed by highly brittle, post-peak
behaviour.
The triaxial testing of the cement properties revealed:
• The peak load was almost independent of the loading rate, within the range considered.
• The maximum strength of specimens increased significantly as the confining pressure
increases.
• The specimens exhibited more ductile behaviour in confined compression tests in
which the gradient of the load-displacement graph tends towards a plateau by the end
of the test.
The effect of curing regime on the mechanical properties of the cement class G showed:
93 |
ADE | Evaluation of Cement Sheath Integrity Reflecting Thermo-Plastic
Behaviour of the Cement in Downhole Conditions (PAPER-3)
ABSTRACT
The cement sheaths play an important role to provide complete zonal isolation during the
wellbores lifetime. The cement sheaths are subjected to pressure and temperature variations which
may lead to different failure mechanisms and subsequently compromising the integrity of the
wellbores. This paper demonstrates the results of three-dimensional finite element frameworks
employing the Concrete Damage Plasticity (CDP) model for the cement sheath and a surface-
based cohesive behaviour along with thermal conduction behaviour at the interfaces to assess the
integrity of cement sheaths subjected to mechanical and thermal loads. The occurrence of the
compression and tensile damage within the eccentric cement sheaths, and also the propensity of
interfaces debonding were investigated considering different wellbore operational scenarios.
Based on the simulations results controlled heating rates might lead to less potential compression
damage. The tensile damage magnitude and its localisation are more dependent on the geometry
of the wellbore instead of the heating rates, and the importance of casing centralisation was
highlighted. The impacts of different cooling scenarios on the cement sheath damage were shown
to be minimal due to the dominant effect of pressurizing the wellbore and the in-situ stresses
confinement.
Keywords: cement sheath integrity, concrete damage plasticity model, wellbores architecture,
thermal rates, compression damage, tensile damage
7.1.Introduction
The exploration and exploitation of hydrocarbon wells should be in agreement with the
protection of the environment to prevent groundwater (aquifers) pollution [14, 15] and migration
of fugitive emissions [16] into the atmosphere [9]. Groundwater sources are protected from the
contents of well operational processes, i.e. drilling, pressure integrity tests (Leak-off tests),
hydraulic fracturing, production operations, etc. by layers of steel casing, and cement sheaths
which perform as multiple barriers to provide complete zonal isolation [17]. Although wellbores
are sealed and block any interaction between formation fluid and geologic strata (which may
contain groundwater), the integrity of wellbores might still be compromised [9, 18]. At this stage,
wellbores may turn into the high-permeability conduits for the formation fluids [19] which induces
a potential risk to the environment by contaminating the groundwater and atmosphere. To maintain
the integrity of the wellbores, a wellbore barrier system should be designed in a way to endure the
100 |
ADE | mechanical and thermal operational procedures applied by the production and recovery phases
during a well lifetime.
Well-cementing (cementation) is an important stage in the wellbore completion procedure as
the cement sheath is responsible for maintaining the integrity of the wellbores [40]. The
permeability of cement used in the oil and gas industry (cement class G) is very low, usually less
than a 0.2 mD [38]. Therefore, the hydraulic isolation is straightforwardly achieved, and any
probable leakage path can be created only through flaws in cement placement procedures or
cement mechanical failures which result in the formation of cracks within the cement sheath /and
formation of micro-annulus at the interfaces of cement sheath with the casing and the rock [11].
The mechanical and thermal stress state of a cement sheath is subjected to pressure and
temperature variations due to different reasons for instances casing expansion/contraction [32],
leak-off tests [23], hydraulic fracturing [33], loading from formation stresses such as tectonic
stress, subsidence and formation creep [12], change of pore pressure or temperature [34], normal
well production [12], injection of hot steam of cold water [35, 36], and cement hydration [146].
These operational procedures have significant effects on the integrity and failure mechanisms of
cement sheaths. The integrity of the cement sheath is also dependent on cement mechanical
properties [31, 44, 45], the cement bond strength [37, 47], cement history (cement shrinkage) [8],
far-field stresses [48], and well architecture (cement sheath thickness, formation properties,
cement sheath eccentricity, and wellbore deviation [8, 37]).
So far, different analytical and numerical modelling approaches were performed to achieve a
better evaluation of cement sheath integrity in wells. The accuracy of analytical models and
consequently their results are limited to the accuracy and suitability of their initial assumptions
and employing simplifications to facilitate finding solutions [47]. On the other hand, numerical
modelling including Finite Element Method (FEM) can be more practical with respect to its ability
to incorporate material non-linearity, different types of geometry and boundary conditions, and
in-situ stress conditions [47]. The reliability of these numerical models is conditional on the
accessibility, validation, and verification of experimental data [47].
Up to now, the linear elastic approach was utilised in a few cement integrity analyses, i.e. [28,
37, 48, 49]. Li, Liu, Wang, Yuan and Qi [48] developed a coupled framework to investigate the
effect of non-uniform in-situ stress filed, temperature, and pressure effects on wellbore integrity.
The stress states evaluated assuming the linear elastic behaviour for all the materials. According
to this study, the anisotropy of in situ stresses resulted in the creation of shear stresses and non-
uniform stress distribution within the cement sheath. By increasing the casing temperature, the
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ADE | tensile stresses develop and lead to the creation of fractures in the inner surface of the cement
sheath.
Guo, Bu and Yan [49] presented a numerical study to investigate the effect of the heating
period, cement thermal expansion, and overburden pressure on the cement integrity under steam
stimulation conditions. All materials presumed to be linear elastic. They recommended a moderate
heating rate and moderate cement thermal expansion coefficient is beneficial to maintain the
cement sheath integrity.
De Andrade and Sangesland [66] conducted a numerical study with a particular focus on
thermal-related load cases. They built a two-dimensional (2-D) model and assumed a linear elastic
behaviour for all the materials, bonded contact between wellbore components and isotropic in-situ
stresses. A utilisation factor based on Mogi-Coulomb criterion was defined to check the state of
the stress and estimate cement sheath failure. The utilisation of Mogi-Coulomb criterion instead
of Mohr-Coulomb was explained by considering the obtained experimental data by Al-Ajmi [67]
which states Mogi-Coulomb criterion represents the state of shear failure in different types of
rocks better than Mohr-Coulomb criterion. According to their results, during heating procedures,
the failure of the cement sheath may occur due to shear stress, and the possibility of debonding
failure during cooling procedures is high. Based on their results, the effect of casing centralisation
and controlled heating/cooling rates seemed to be trivial.
Roy, Morris, Walsh, Iyer, Carroll, Todorovic, Gawel and Torsæter [60] carried out an
experimental-numerical study to investigate the effect of wellbore size, cement Young’s modulus,
and different cooling rates on the imposed thermal stresses during CO injections. They coupled a
2
finite element solver assuming linear elastic materials with a finite volume heat equation solver to
simulate the mechanical response of the materials exposed to thermal loading. Their observations
showed the variation of cement Young’s modulus effects the magnitude of maximum radial stress
at the interface of cement and casing or the interface of cement with the formation. They also
showed the overall stress within different wellbore components is more dependent on the
temperature gradient rather than the temperature difference between the initial and ultimate state
in materials.
Although the linear elastic approach has been used in some studies including the
aforementioned studies, the obtained stress-strain curves from the isotropic drained compression
tests on the cementitious specimens by [36, 72] are shown to be non-linear. Therefore, the
employment of linear elastic theory in cement integrity simulations troubles the accuracy and
reliability of the results. Additionally, the existence of the permanent strains upon unloading [36]
102 |
ADE | confirms the incompatibility of linear elastic theory in cement integrity evaluations again as the
elastic theory does not incorporate the time-dependency and materials hysteresis law [50].
The non-linear approaches including those using Von-Mises [51], Ottosen model [5, 53],
Drucker-Prager [52], modified Cam-Clay [36], and Mohr-Coulomb / with smeared cracking [31,
44] were incorporated in the cement sheath integrity evaluations to alleviate the shortcomings of
the linear elastic models.
Fleckenstein, Eustes and Miller [51] employed the von-Mises criteria and showed that the
magnitude of tangential stresses would be significantly reduced if the cement sheath acts as a
ductile material with lower Young’s modulus and higher Poisson’s ratio. The lack of pressure
dependency of the von Mises criteria is however problematic in modelling cementitious materials.
Asamoto, Le Guen, Poupard and Capra [5], Guen, Asamoto, Houdu and Poupard [53]
developed a 2-D model using the Ottosen model [71] as a smeared crack model to investigate the
softening post-peak behaviour of the cement sheath and the estimation of the crack width in a
wellbore subjected to thermal and mechanical loads. Guen, Asamoto, Houdu and Poupard [53]
examined the effect of temperature and pressure changes on the thermo-mechanical response of a
wellbore for the application to Ketzin injection well using FEM. The interfaces of cement-casing
and cement-rock were modelled using joint elements with Mohr-Coulomb failure criterion. They
concluded that the possibility of debonding as a result of CO injections is very low, except in the
2
scenarios with a very high degree of eccentricity (85% eccentricity) in which the cement tangential
stress would exceed the cement sheath tensile limit. In both studies, the details of the constitutive
model performance and the relevance of the constitutive parameters to the experimental data are
not described.
Zhang, Yan, Yang and Zhao [52] developed an analytical plain-strain model to evaluate the
integrity of a wellbore under HPHT conditions coupling displacement and temperature approach.
The Mises criterion, Drucker-Prager, and Joint Roughness Coefficient-Joint wall Compressive
Strength (JRC-JCS) were utilised to model the casing, cement sheath and cement interfaces
respectively. According to their parametric study, using cement with low Young’s modulus and
high Poisson’s ratio improves wellbore ability to maintain its integrity. However, the studies
carried out on the performance of Drucker-Prager model shows this model does not provide
accurate predictions while one or more principle stresses are tensile stress. Additionally,
considering the same effect for 𝜎 and 𝜎 leads to overestimation of rocks’ strength and it is not
2 3
verified by laboratory experimental data [69, 70].
103 |
ADE | The modified Cam-Clay model has been suggested as a method to incorporate cement micro
cracking mechanisms by Bois, Garnier, Rodot, Sain-Marc and Aimard [36] owing to the
nonlinearity of stress-strain curve achieved from the isotropic drained compression tests [72] and
heterogeneous nature of cement at the microscale. Although important aspects of materials
behaviour (material strength, compression or dilatancy, and critical state of elements under high
distortion) are considered in this model, the tensile post-peak material is not incorporated into this
framework.
Bosma, Ravi, van Driel and Schreppers [44] developed a two-dimensional (2-D) model
incorporating Mohr-Coulomb plasticity combined with smeared cracking to model the cement
sheath under compression/shear and tension along with heat transfer phenomena. The cement
sheath interfaces were modelled using interface elements applying a coulomb friction criterion.
Their results showed that cement sheath failure is happened because of shear stresses caused by
in situ stresses or either due to tensile failure mechanisms which is more likely when the cement
Young’s modulus is higher than the rock.
Ravi, Bosma and Gastebled [31] developed a 2-D model to investigate the wellbores integrity
subjected to operational procedures. To model the stress state within the cement sheath, the
Hookean model was incorporated for undamaged state and combined Mohr-Coulomb plasticity
with smeared cracking after exceeding the compressive shear and tensile strength state. According
to their findings, the integrity of the cement sheath is highly dependent on the cement and
mechanical rock properties, and well-operating parameters. Moreover, cement sheath with less
stiffness shows more resilient and helps to reduce the risk of cement sheath failure.
Mohr-Coulomb criterion was also used by Feng, Podnos and Gray [50], Nygaard, Salehi,
Weideman and Lavoie [74], and Zhu, Deng, Zhao, Zhao, Liu and Wang [75] to predict the plastic
behaviour of the cement sheaths subjected to mechanical and thermal loads.
The combination of Mohr-Coulomb with smeared cracking is one of a few suitable approaches
for modelling the real conditions in the cement integrity numerical simulations. However, despite
the broad application of Mohr-Coulomb criteria, it has its own limitations. The model assumes a
linear relationship between √J and I in the meridian plane, while this relationship has been
2 1
experimentally shown to be curved [36, 72, 76, 77], for cementitious materials, particularly at low
confinement. The major principal stress 𝜎 and intermediate principal stress 𝜎 are defined
1 2
independently in Mohr-Coulomb model which results in an underestimation of the yield strength
of the material and, it is not in a good agreement with experiments in which the effect of 𝜎 is
2
being considered. The shape of the yield surface in the deviatoric plane is an asymmetrical
104 |
ADE | hexagon, whereby the sharp corners can hinder convergence in numerical simulations [70, 78].
Moreover, quasi-brittle materials experience a huge volume change due to a large amount of
inelastic strains (dilatancy) which has been overlooked so far by using associated flow rules in the
aforementioned modelling approaches of the cement sheath. The associative plastic flow rules
tend to lead to poor results in dilatancy evolution [55].
The application of plasticity theory in compression (Mohr-Coulomb) combined with the
fracture mechanics models such as smeared cracking presents some drawbacks as well. Given
that, smeared crack models in finite element analysis can often be problematic in terms of “mesh
alignment sensitivity” or “mesh orientation bias” which indicates that the orientation of smeared
crack depends on the discretization orientation [79]. It is worth adding that the mesh regularization
approach proposed by [56] (crack band theory) in the smeared cracking model has been successful
for predicting mode I fractures while the extension of this approach to mixed-mode failure and
three-dimensional stress state is hard [79].
Wellbore integrity modelling has been significantly progressed regarding complexity and
capacity to assess the integrity of wellbore barriers. However, some aspects of wellbore integrity
modelling still require improvements, in particular, the incorporation of appropriate cement
constitutive law. The softening aspects of constitutive models, and subsequently, the evolution of
corresponding cement constitutive parameters requires more attention to achieve a reliable and
efficient model. Considering these limitations, it would be practical to employ more suitable
models with respect to their accuracy (enrichment) and reliability (capability to reproduce the
experimental data) along with their efficiency (mesh orientation and mesh size objectivity) [79].
In this paper, the Concrete Damage Plasticity model (CDP) specifically formulated for the
modelling of geo-materials developed and modified by [54, 55] is utilised. The major advantage
of Concrete Damage Plasticity (CDP) model is coupling plasticity with damage mechanism which
explains the elastic stiffness degradation of materials clearly during the experiments due to the
formation of microcracking. The formation of microcracks which is also characterized as
softening behaviour of the materials is difficult to describe applying classical plasticity models
[55]. The modified version of CDP by [55] benefits from considering the difference in tensile and
compressive responses of geo-materials as these materials undergo different states of damage
while being subjected to different loading conditions. This model also takes into account the
materials pressure-dependency behaviour under shearing at different levels of confinement. The
non-associated flow rule which represents the dilatancy of the geo-materials is also embedded into
this model. These features make this model a very suitable model to be applied to the range of
105 |
ADE | geo-materials including rocks, and cement-based materials [55] compared to the rest of model
used . The corresponding CDP model parameters were obtained from the previous experimental-
numerical study on cement class G by [116, 147].
In this paper, the susceptibility and magnitude of compression damage, tensile damage, and
interfaces debonding in the cross-sectional slices of two wellbores are investigated considering
different well-operating scenarios. The parametric study is carried out to assess the effect of
wellbore architecture (e.g. eccentricity, different layers of cement sheath and casing), different
heating and cooling rates on the integrity of cement sheath.
This paper is organised as follows; section 7.2 describes an overview of the finite element
frameworks including wellbore geometries, initial states of the stresses, the mechanical and
thermal behaviour of interfaces modelling, and material properties (i.e. cement constitutive
modelling). The effect of enhanced pressure and temperature on wellbore-1 is investigated in
section 7.3. Section 7.4 and 7.5 investigate the effect of heating and cooling scenarios along with
an applied pressure respectively. The propensity of compression and tensile damage with respect
to heating and cooling scenarios are examined and also compared in these two sections. Section
7.6 describes the susceptibility of forming micro-annuli at the interfaces of cement sheaths in
wellbore-2 due to the heating and cooling operational procedures followed by conclusion in
section 7.7.
7.2. Overview of Finite Element Modelling
Three-dimensional finite element frameworks were developed to investigate the effect of
pressure and temperature variations events on the cement sheath integrity using ABAQUS /
Standard software package. 8-node thermally coupled brick, trilinear displacement and
temperature elements were utilised to mesh the system components. A fully coupled transient
thermal-stress analyses procedures were undertaken. Different scenarios were chosen to
investigate the effect of different well operational procedures in completion stages. Figure 7.1
demonstrates wellbore schematic sections. The information regarding wellbore-1 is based on an
actual site on Ketzin, Germany [5, 148], while, wellbore-2 is a case study selected to examine the
effect of different thermo-mechanical loading scenarios, and wellbore architecture (different
degrees of eccentricity) on the cement sheaths integrity.
106 |
ADE | The heating and cooling scenarios represent the different operational procedures leading to
pressure and temperature variations applied to wellbore-2. For instances, during the
commencement of production procedures, the pressure and temperature are increased within the
wellbores to enforce the hydrocarbon flow from the reservoir [66], and in the start of injection
procedures, wellbores are cooled down for the fluid to flow into the reservoir [66].
Wellbore-1
Casing
Cement Sheaths
Wellbore-2
Casing
Cement Sheath
Figure 7.1: Wellbores Schematic Sections after [5]
In Abaqus/standard the temperatures are integrated utilising a backwards-difference scheme,
and non-linear coupled system is solved using Newton’s method. The exact implementation of
Newton’s method for fully coupled temperature-displacement was applied involving a non-
symmetric Jacobian matrix as shown in the following equation [57].
𝐾 𝐾 ∆𝑢 𝑅
[ 𝑢𝑢 𝑢𝜃]{ }= { 𝑢} (7.1)
𝐾 𝐾 ∆𝜃 𝑅
𝜃𝑢 𝜃𝜃 𝜃
where ∆𝑢 and ∆𝜃 are the respective corrections to the incremental displacement and temperature,
𝐾 are submatrices of the fully coupled Jacobian matrix, and 𝑅 and 𝑅 are the mathematical and
𝑖𝑗 𝑢 𝜃
thermal residual vectors respectively.
Unsymmetrical matrix storage and solution scheme should be undertaken to solve the system
equations. The mechanical and thermal equations must be solved simultaneously.
The governing equations are as follows [149].
Kinematic relation:
107 |
ADE | 1
𝜀 = (𝑢 +𝑢 ) (7.2)
𝑖,𝑗 2 𝑖,𝑗 𝑗,𝑖
Motion equation:
𝜎 +𝜌𝐹 = 𝜌𝑢̈ , 𝜎 = 𝜎 (7.3)
𝑖𝑗,𝑗 𝑖 𝑖 𝑖,𝑗 𝑗,𝑖
where 𝜌 is the mas density, and 𝐹 is external force per unit mass, .
𝑖
Energy-scale equation:
𝑞 +𝜌(𝑇 𝑠̇ −𝑅)= 0 (7.4)
𝑖,𝑖 𝑜
where 𝑞 is the heat flux per unit area, 𝑇 is the initial temperature, s is entropy per unit mass, and
𝑖 𝑜
R is internal heat capacitance per unit mass.
Constitutive equations:
𝜎 =𝐶 𝜀 +𝛽𝜃 (7.5)
𝑖,𝑗 𝑖,𝑗𝑘𝑙 𝑘𝑙
where 𝐶 is the stiffness tensor, 𝛽 is thermal stress, and 𝜃 is the temperature difference.
𝑖,𝑗𝑘𝑙
𝜌𝑐 𝐸𝛼
𝑞 = −𝑘 𝜃 and 𝜌𝑠 = 𝜃−𝛽𝜀 , 𝛽 = (7.6)
𝑖 𝑖,𝑗 ,𝑗 𝑇0 𝑖,𝑗 (1−2𝜗)
where 𝑘 is the thermal conductivity tensor, c specific heat per unit mass at constant strain, 𝑇 is
𝑖,𝑗 0
the initial temperature, 𝐸 and 𝜗 are Young’s modulus and Poisson’s ratio respectively, and 𝛼 is
coefficient of linear thermal expansion.
From the equation of motion (7.2) and the energy-scale equation (7.4) using the constitutive
equations (7.5) -(7.6) the general basic equations will be obtained as follows.
(𝐶 𝜀 ) +(𝛽𝜃) +𝜌𝐹 −𝜌𝑢̈ = 0 (7.7)
𝑖,𝑗𝑘𝑙 𝑘𝑙 ,𝑗 ,𝑗 𝑖 𝑖
(𝑘 𝜃 ) +𝜌𝑐𝜃̇ +𝜌𝑅−𝑇 𝛽𝜀̇ = 0 (7.8)
𝑖,𝑗 ,𝑗 ,𝑖 0 𝑖,𝑗
7.2.1. Initial State of Stress and Boundary Conditions
The anisotropic geo-stress components (𝜎 and 𝜎 ) were applied in the initial step of the
𝐻 ℎ
analyses. The geo-stresses information were extracted from a geo-mechanical study performed by
Ouellet, Bérard, Desroches, Frykman, Welsh, Minton, Pamukcu, Hurter and Schmidt-
Hattenberger [148] on the Ketzin site. The anisotropy of geo-stresses would cause further shear
stresses to the wellbore [82] and is required to be considered in cement integrity modelling. An
kPa
overburden stress gradient of 22.6 was assumed. The ratio of maximum horizontal stress to
m
108 |
ADE | overburden stress
(σH)
and the ratio of ansitropic in-situ stresses
(σH)
is assumed to be 0.8 and
σV σV
0.7 respectively. The variation of overburden (vertical) initial stress in depth is negligible since
the ratio of model height to the width is comparatively small. The corresponding overburden
effective stress at the casing shoes were computed according to the located depth and formation
density, and all shear components are assumed to be zero. The displacement constraints were
applied to the normal direction of the bottom surface, the outer surface of the formation, and the
symmetric surfaces.
7.2.2. Interface Modelling
The cement sheath interfaces with the casing and the formation are recognised as the weakest
components to provide an effective leakage barrier [37, 81, 150]. The mechanical behaviour and
failure of the interfaces can be described by a cohesive model for interfaces between two different
materials [151]. In this study, the mechanical behaviour of cement sheath interfaces is represented
by surface-based cohesive behaviour accompanied by the thermal interaction properties to model
heat conduction at the interfaces.
7.2.2.1. Mechanical Behaviour of the Interfaces
The mechanical behaviour of the cement sheath interfaces is modelled by defining surface-
based cohesive behaviour as a surface interaction property with traction-separation capacity.
Traction-separation (t−δ) law can be expressed by different relationships for various materials,
and according to the studies by Hillerborg, Modéer and Petersson [131] bilinear or triangular
traction separation law had successful applications on brittle materials such as cementitious
𝑡
𝑛
(𝑡 , 𝑡 ) Damage Initiation
𝑡0(𝑡0,𝑡0)
𝑛 𝑠 𝑛
Damage Evolution
𝐾0(𝐾0 , 𝐾0)
𝑛 𝑠 𝑡
𝐺𝐶
𝛿 (𝛿 ,𝛿 )
𝑛 𝑠 𝑡
𝛿0 (𝛿0,𝛿0)
𝑛 𝑠 𝑡
No separation Delamination Zone
Figure 7.2: Linear Softening Traction-Separation Law
materials. Figure 7.2 shows the triangular traction separation law.
109 |
ADE | where the superscript 0 denotes the maximum traction or initiation traction value, n,s,t are
representing the normal and shear directions respectively, and K represents the contact stiffness.
The dashed-line demonstrates the stiffness degradation after the peak. Equation (7.9) describes a
linear relationship traction-separation (t−δ) law.
𝑡 𝐾 𝐾 𝐾 𝛿
𝑛 𝑛𝑛 𝑛𝑠 𝑛𝑡 𝑛
{𝑡} = {𝑡 }= [𝐾 𝐾 𝐾 ]{𝛿 }= 𝐾{𝛿} (7.9)
𝑠 𝑠𝑛 𝑠𝑠 𝑠𝑡 𝑠
𝑡 𝐾 𝐾 𝐾 𝛿
𝑡 𝑡𝑛 𝑡𝑠 𝑡𝑡 𝑡
The damage mechanism is defined based on damage initiation criterion and damage evolution
law. The damage initiation between two different materials is usually described by mixed modes
condition [152] via using quadratic nominal stress criterion.
Damage is assumed to initiate once a quadratic interaction function relating the nominal stress
ratios (as defined in the expression below) reaches one. This criterion can be defined as:
2 2 2
〈𝑡 〉 𝑡 𝑡
𝑛 𝑠 𝑡
{ } +{ } +{ } = 1 (7.10)
𝑡0 𝑡0 𝑡0
𝑛 𝑠 𝑡
where 𝑡 , 𝑡 , and 𝑡 are the normal and shear tractions across the interfaces and the superscript 0
𝑛 𝑠 𝑡
denotes the maximum traction or initiation traction value.
Damage evolution law describes the process of failure and the rate at which the material
stiffness is degraded when the corresponding initiation criterion is met. To study the mixed-mode
condition, the Benzeggagh-Kenane (BK) [133] fracture energy criterion is used here with the
assumption that the critical fracture energy during separation along the first and the second shear
direction are the same; 𝐺𝐶 = 𝐺𝐶.
𝑠 𝑡
𝐺 𝜂
𝐺𝐶 +(𝐺𝐶 −𝐺𝐶){ 𝑆 } =𝐺𝐶 (7.11)
𝑛 𝑠 𝑛 𝐺
𝑇
where 𝐺 =𝐺 +𝐺 , 𝐺 = 𝐺 +𝐺 and 𝜂 is a cohesive property parameter [82, 87, 134].
𝑆 𝑠 𝑡 𝑇 𝑛 𝑆
Determination of Cohesive Model Parameters
Carter and Evans [84], Evans and Carter [86] designed a push-out test setup to measure cement
shear bond and hydraulic bond. Shear bond is essential to support the pipe mechanically, whereas
the hydraulic bond prevents the formation of micro-annuli. They designed cylindrical chambers
in which shear bond is determined by applying force to instigate the movements of the pipe
surrounded by cement. The shear bond measured as dividing the force to the contact surface area.
The cement hydraulic bond was defined as the cement bond to the casing or the formation which
prevent the fluid migration. Hydraulic bond was determined by applying pressure to the cement
interfaces until leakage happens. Ladva, Craster, Jones, Goldsmith and Scott [85] repeated the
above experimental procedures using cement class G to measure the shear bonding between
110 |
ADE | cement and formation. Afterwards, Wang and Taleghani [37] performed inverse analyses on the
experimental results of [84-86] to determine the cohesive parameters. Table 7.1 summarised the
obtained cohesive parameters by Wang and Taleghani [37] adopted in this study to model the
mechanical behaviour of the interfaces.
Table 7.1: Cohesive properties of cement / casing and cement/rock [37]
Cohesive Properties Shear Normal Cohesive Critical
Interfaces Strength (kPa) Strength (kPa) Stiffness (kPa) energy (𝑱/𝒎𝟐)
Casing/Cement Interface 2000 500 30𝐸6 100
Cement/Formation
420 420 30𝐸6 100
Interface
Although the cohesive parameters have been determined from a number of experimental
studies, uncertainty still remains in the utilisation of these parameters. Since as stated by Carter
and Evans [84], Evans and Carter [86]’s studies the cement shear bond to the casing is dependent
on curing temperature, the pipe condition, and variations of different cement brands. The
adherence degree of well cement to rock is highly variable and site dependent. The cement
hydraulic bond to the casing and formation is dependent on the type of the formation, the surface
finish of the pipe, type of mud layer, and degree of mud removal [85, 135].
7.2.2.2. Thermal Conduction Behaviour of the Interfaces
The defined thermal contact properties at the interfaces allow the conductive heat transfer
between the surfaces. The thermal conductivity at the contact surfaces is formulated using
Equation (7.12) :
𝑞 = 𝑘(𝑇 −𝑇 ) (7.12)
𝐴 𝐵
where q is the heat flux per unit area crossing the interface from node A on the slave surface to
node B on the opposite surface (master surface), k is the gap conductance, and T and T are the
A B
temperatures of the nodes in the contact surfaces [153].
111 |
ADE | The gap conductance coefficient decreases linearly as the clearance increases due to the
creation of gaps and flaws between the contact surfaces as shown in Figure 7.3. The effect of
surrounding temperature on the gap conductance coefficient is not seen in this study also the
Figure 7.3: Gap Conductance vs. Separation between Two Surfaces in Contact
thermal contact resistance is assumed to negligible and complete heat transfer conditions across
the contact surfaces modelled by defining significantly high thermal contact conductance
coefficient at the beginning of analysis at which the contact surfaces are fully bonded. The
combination of surface-based cohesive behaviour and thermal conductance behaviour has been
used in some studies of thermo-mechanical damage modelling composites by different authours
(e.g. [134, 154]) but not in this field to the best of our knowledge.
7.2.3. Material Properties
The behaviour and failure mechanism of the cement sheath was modelled by using Concrete
Damage Plasticity (CDP) model and calibrated according to numerical-experimental studies
performed on cement class G by [116, 147]. Elastic mechanical properties of the steel casing and
different rock formations are defined as shown in Table 7.2 after [5].
Table 7.2: Mechanical Properties of Casing and Rock Formation [5]
Thermal Thermal
Specific
Young’s
Conductivit Heat Expansion Density Poisson’s
Materials y 𝐽 Coefficient ( 𝑘𝑔 ⁄ ) Modulus Ratio
(𝑊 ⁄ 𝑚.𝐾) ( ⁄ 𝑘𝑔.𝐾) ( μ ⁄𝑜 𝐶) 𝑚3 (𝐺𝑃𝑎)
Casing 54 470 12 7850 200 0.3
Formation
Wellbore-
4.5 1000 10 3000 20 0.29
1
Wellbore-
3.0 1500 13 2300 5.5 0.3
2
112 |
ADE | 7.2.3.1. Cement Constitutive Modelling
The observed non-linearity in obtained stress-strain curve studying cement mechanical
behaviour under compression tests [10, 55] results from two different microstructural changes
which happen in the materials while subjected to different loading conditions. One is plastic flow
causes the permanent deformation and the second is the development of microcracks which leads
to elastic stiffness degradation [56]. Therefore, it is necessary to apply a model which combines
plasticity and damage mechanics. The creation of microcracks which is also characterized as
softening behaviour of the materials is difficult to explain using classical plasticity models [31].
The damage mechanism is described by two physical aspects corresponding to the two modes of
cracking (hardening and softening) [56].
Therefore, in this study, Concrete Damage Plasticity (CDP) model developed by [54] and then
modified by [55, 115] has been employed. The Concrete damage plasticity (CDP) model is a
continuum model based on damage mechanics and plasticity theory which can be used in
improving the prediction of cement class G the behaviour, from initial yield to failure. In the
modified revision, two damage variables one for compressive damage and one for tensile damage
were incorporated to consider different states of damage. This feature makes the model capable of
describing the induced anisotropy of microcracking which also facilitates the numerical
implementation procedures [55, 115]. The pressure-sensitive yield criterion accompanied by
employing the dilatancy (non-associated flow rule), makes this model more suitable than the
others that have been employed in the assessments of cement sheath integrity.
The uniaxial tension response is characterised by a linear elastic relationship until reaching the
failure stress (𝜎 ) which corresponds to the beginning of micro-cracking in the material. Beyond
𝑡
the failure stress, the effects of micro-cracking are taken into account in the model using a
softening stress-strain response. The uniaxial compression response is also characterised by a
linear elastic relationship until reaching the initial compressive strength (𝜎 ) followed by stress
𝑐
hardening in the plastic region up to the ultimate stress (𝜎 ). Strain softening occurs subsequent
𝑐𝑢
to reaching the ultimate stress.
The stress-strain relations under uniaxial tension and compression in the CDP model are defined
as follows respectively.
𝜎 = (1−𝑑 )𝐸 (𝜀 −𝜀̃𝑡 ) (7.13)
𝑡 𝑡 0 𝑡 𝑝𝑙
𝜎 = (1−𝑑 )𝐸 (𝜀 −𝜀̃𝑐 ) (7.14)
𝑐 𝑐 0 𝑐 𝑝𝑙
where d and d are tensile and compression damage variables; E is initial undamaged
t c 0
stiffness; ε̃t , ε̃c are tensile and compressive equivalent plastic strains respectively.
pl pl
113 |
ADE | In this study, the compression damage d was computed using Equation (7.15) [4] as shown in
c
Figure 7.4:
d = 1−
σ c,
(7.15)
c
σcu
where 𝜎 ′ is the axial stress of the cement on the descending branch, and 𝜎 is the peak point of
𝑐 𝑐𝑢
the stress-strain curve.
Tensile damage d was defined using a linear relationship [131] between cement tensile strength
t
and cracking displacement as shown in Figure 7.5.
Figure 7.4: Compression Damage vs. Inelastic Strain Figure 7.5: Tensile Damage vs. Cracking Displacement
The shape of yield surface in the deviatoric plane changes according to the ratio of the second
stress invariant on the tensile meridian to the compressive meridian which allows capturing the
material behaviour very well. This yield function was defined by Lubliner, Oliver, Oller and Onate
[54] with some modifications made by Lee and Fenves [55], Lee and Fenves [115] afterwards to
interpret the evolution of strength under tension and compression. It is defined as follows.
1
𝐹 = 1−𝛼(𝑞̅−3.𝛼.𝑝̅+𝛽(𝜀 𝑝̃ 𝑙)〈𝜎̅ 𝑚𝑎𝑥〉−𝛾〈−𝜎̅ 𝑚𝑎𝑥〉)−𝜎̅ 𝑐(𝜀 𝑝̃𝑐 𝑙)= 0 (7.16)
1
where 〈 〉 is the Macaulay bracket, 𝜎̅ is the maximum principle effective stress, 𝑝̅ = − 𝜎̅ .𝐼
𝑚𝑎𝑥
3
is the effective hydrostatic stress and 𝑞̅ = √3 𝑆̅.𝑆̅ is the Mises equivalent effective stress with 𝑆̅ =
2
𝑝̅𝐼+𝜎̅ being the deviatoric part of the effective stress tensor. The function 𝛽(𝜀̃ ) in (7.8) is
𝑝𝑙
defined as follows.
𝜎̅ (𝜀̃𝑐 )
𝑐 𝑝𝑙
𝛽(𝜀̃ )= (1−𝛼)−(1+𝛼) (7.17)
𝑝𝑙 𝜎̅ (𝜀̃𝑡 )
𝑡 𝑝𝑙
in which two cohesion stresses are employed for the modelling of cyclic behaviour.
114 |
ADE | 𝜎
( 𝑏𝑜)−1
𝜎
𝛼 = 𝑐 (7.18)
𝜎
2( 𝑏𝑜)−1
𝜎
𝑐
𝜎
where 𝑏𝑜 is the ratio of biaxial compressive yield stress to uniaxial compressive yield stress. The
𝜎𝑐
shape of loading surface in the deviatoric plane is controlled by parameter 𝛾 in Equation (7.11)
[123] and define as
3(1−𝐾 )
𝑐
𝛾 = (7.19)
2𝐾 +3
𝑐
where 𝐾 =
(√𝐽2)𝑇𝑀
is a coefficient determined at a given state 𝑝̅ , 𝐽 is the second invariant of
𝑐 2
(√𝐽2)𝐶𝑀
stress with the subscripts TM and CM employed for the tensile and compressive meridians
respectively and must satisfy the condition 0.5 ≤ 𝐾 ≤ 1 . For the non-associated flow rule, the
𝑐
plastic potential 𝐺 in the form of the Drucker-Prager hyperbolic function is used.
𝐺 = √(𝜖𝜎 tan𝜓)2+𝑞̅2−𝑝̅.tan𝜓 (7.20)
𝑡𝑜
In which 𝜎 is the uniaxial tensile stress at failure, the dilation angle ψ is measured in a p-q plane
𝑡𝑜
at high confining pressure, and 𝜖 is an indicator for the eccentricity of the plastic potential surface.
Determination of Cement Constitutive Model Parameters
The selection of an appropriate constitutive law for the cement sheath as a geo-material and its
corresponding model parameters are the utmost of importance part of wellbore integrity
modelling. The constitutive parameters were determined through performing experiments and the
interpretation of experimental results to obtain the failure envelope of the constitutive model
(Concrete Damage Plasticity Model). The experimental outcomes of uniaxial, triaxial
compression tests and three-point bending tests performed by Arjomand, Bennett and Nguyen
[116] (also completely explained in Chapter 6) on cement class G specimens were utilised in this
study. The experimental set-up and procedures were briefly described in the following section.
The cylindrical and prismatic specimens were cured in a water tank with an automatic
thermostat was set on 30oC for 28 days. Prior to performing compression tests the surface of the
cylindrical samples was ground to achieve smooth surfaces in the way that the ends were
completely orthogonal to the cylinder’s longitudinal axis [100]. For this purpose, the sample
moulds were designed 3 mm taller than the desired sample length. The uniaxial compression tests
were run with the displacement rate of to 0.04 mm/min, at which rate the specimens showed less
brittle behaviour. Triaxial compression tests were performed with confining pressures (P) of 15
c
MPa and 30 MPa. The loading path was designed so that the pressure confinement reached the
desired confining pressure at the first step and then were loaded axially under displacement control
115 |
ADE | until failure occurred. Three-point bending tests on notched and un-notched beam samples were
performed to obtain the cement class G tensile strength and fracture energy accompanied by
applying modifications on three-point bending set-up. The approximate shape of the yield surface
for concrete damage plasticity models and the corresponding constitutive parameters were
obtained through parameters calibration as described in [147] and chapter 6 of this thesis.
The cement sheath thermal properties were taken from the study performed by Asamoto, Le
Guen, Poupard and Capra [5]. Table 7.3 summarises the cement mechanical and thermal properties
utilised in this study.
Table 7.3: Cement Thermal [5] and Mechanical Properties [147]
Cement Thermal Properties
Thermal Specific Thermal Expansion Coefficient
Conductivity Heat μ
( ⁄ )
(𝑾
⁄ )
(𝑱
⁄ )
𝑶𝑪
𝒎.𝑲 𝒌𝒈.𝑲
1.2 2100 9
Cement Mechanical Properties
Young’s Poisson’s Density
Dilation angle Eccentricity
modulus ratio (υ) 𝒌𝒈
( ⁄ ) 𝝍 (degrees) 𝜺
𝑬 (GPa) 𝒎𝟑
𝟎
6.8 0.25 1900 42 0.1
Initial Ultimate Tensile stress
𝝈 𝒃𝒐 𝑲 𝒄 compressive compressive 𝝈 𝒕 (MPa)
𝝈 stress stress
𝒄
𝝈 (MPa) 𝝈 (MPa)
𝒄 𝒄𝒖
1.16 0.8 50 55 1.92
7.3.Influence of Enhanced Pressure and Temperature on Wellbore-1
The data pertaining to different wellbores geometries, casing and formation material properties
are extracted from a case study on the Ketzin site by [5, 53, 148]. A wellbore diameter is a few
tens of centimetres whereas the wellbores depth can reach 5000 m, consequently, modelling a
wellbore in field scale consisting of surrounding formations requires a very long computational
time and excessive finite elements [5]. Therefore, in this study cross-sectional cuts with the height
of five inches (0.127 meter) and half of the model due to the symmetric aspect were considered
for modelling purposes.
The shallower wellbore (wellbore-1) consists of two layers of casing and cement sheaths located
at a shallow depth of 600 m surrounded by caprock. Wellbore-1 was encased by two layers of
1" 5"
conductor and surface casings with the diameter of 5 and 9 and thicknesses of 9.2 mm and 8.9
2 8
mm, respectively. To perform more realistic simulations, stress-related factors which induce
116 |
ADE | wellbore failure in some fields were incorporated within the frameworks including employing
anisotropic in-situ stresses as stated in section 7.2.1, and 50% eccentricity applied to the layers of
the cement sheath.
Figure 7.6 shows different geometries considered for wellbore-1. The effect of model size
including the surrounding formation is important to prevent any artificial effects in the stress
distributions and to assure that far-field stresses are applied from a reasonable distance from the
wellbores [155]. Salehi [114] suggested that the model size is better to be at least four times bigger
Figure 7.6: Wellbore-1 with Three Different Geometries (Concentric and Eccentric)
than the borehole size. Furthermore, the element size within the section near the wellbore should
be smaller than the rest of the formation. This finer section should be at least 2-3 times bigger than
the borehole size to improve accuracy [114]. Therefore, the formation rock was partitioned into
two sections and meshed with finer mesh near the wellbore area and coarser mesh in the far field
area.
The degree of eccentricity (ɛ) can be computed according to Equation (7.21) [156].
δ
ɛ %= ×100 (7.21)
(Rw−rc)
where δ is the distance between the casing centre from the wellbore centre, R is the wellbore
w
radius, and r is the casing radius. The degree of eccentricity varies from 0 which is completely
c
centralised casing to 100% which means the outer wall of the casing touches the inner of the
cement sheath and cement sheath thickness is zero at one side [157].
Wellbore-1 is subjected to CO injection conditions imposed to the inner surface of conductor
2
casing while the injection pressure is 7.5 MPa and the maximum CO temperature is set to 310.15
2
K. The initial temperature (T ) is assumed to be constant for the entire wellbore and surrounding
0
formation with the initial temperature of 305.15 K [53]. Anisotropic in-situ stresses were applied
to the model (section 7.2.1) using some of the magnitudes taken from geological information study
on the Ketzin site [148].
117 |
ADE | compression and tensile damage. The obtained results are in good agreement with the other studies
carried out on this case-study, for instance, Asamoto, Le Guen, Poupard and Capra [5]. However,
wellbore-1 was placed in a critical location due to its vicinity to the underground water. Therefore,
the injection of higher pressure and temperature variations in this location should be executed with
caution.
7.4.Influence of Heating Scenarios Operated along with Pressure on Wellbore-2
The deeper wellbore (wellbore-2) consists of single intermediate 7"steel casing thickness of 9
mm and one layer of cement sheath surrounded by unconsolidated sandstone formation located at
a depth of 1000 m. Three different degrees of eccentricity of 30%, 50% and 70% were assessed
for wellbore-2 as shown in Figure 7.8.
Figure 7.8: Wellbore-2 with Three Different Degrees of Eccentricity (30%, 50%, and 70%)
The contribution of thermal loading scenarios along with different heating rates on cement
sheath stress state, plastic deformations, and debonding within the cement sheath corresponding
to the different mechanical-thermal scenarios and three different degrees of eccentricity were
analysed. The materials' mechanical-thermal properties, the magnitude and arrangements of
anisotropic in-situ stresses are maintained constant for all the following analyses.
Wellbore-2 is subjected to 18 MPa pressure at the inner wall of the casing along with
temperature variations. The initial temperature (T ) is assumed to be constant for all the model
0
components as 303.15 K. Temperature variations of ∆T = (T −T )= (573.15 K−
1 0
303.15 K) considering three different heating rates were applied at the inner wall of the casing.
7.4.1. Compression Damage Considering Heating Scenarios
The potential crushing (d ) occurrence caused by pressuring the wellbore, temperature changes
c
and employment of anisotropic in-situ stresses within the cement sheaths are examined through
the local compression damage contours and a global compression damage indicator in the
following sections. Figure 7.9 illustrates the local compression damage contours within the cement
sheaths subjected to pressure and temperature changes with three different heating rates of instant
119 |
ADE | 1.2oC 0.5oC
heating, and controlled heating rates of and . All the damage contours were scaled from
min min
zero to one for comparison purposes.
Figure 7.9: Local Compression Damage Contours within the Cement Sheaths Subjected to Pressure during Heating
Scenarios
The additional shear stress caused by the anisotropy of in-situ stresses and casing expansion
due to the imposed thermal loads resulted in the creation of radial cracks within the cement sheaths
as can be seen in Figure 7.9. The compression damage within the wellbores is more distributed in
wellbores with 30% and 50% eccentricity. While for the wellbore with 70% eccentricity the
compression damage is highly concentrated in the narrower parts. The local maximum
compression damage occurred in the narrower part of cement sheath for all the cases regardless
of mechanical and thermal loading scenarios which shows the dominant effect of eccentricity on
the stress distribution within the cement sheaths. The highest magnitude of local compression
damage (d =0.95) occurred within the cement sheath with 70% eccentricity subjected to instant
c
heating. Whereas, this magnitude reduces to 0.47 in wellbores subjected to controlled heating rates
for the same degree of the eccentricity. This pattern repeated for the wellbores with 50% and 30%
120 |
ADE | eccentricity (the maximum local compression damage occurred within the wellbores subjected to
instant heating). The considerable difference in maximum compression damage magnitude is
indicative of the destructive impact of instant heating on causing crushing damage within the
cement sheaths.
Figure 7.10 shows the temperature gradients within the casing and the 70% eccentric cement
sheath after subjected to different heating rates. As can be seen in Figure 7.10, by the end of the
simulation time, the temperature was consistent throughout the entire casing while the cement
sheath experienced a temperature gradient across the whole section. A significant thermal gradient
was also noticeable from the narrow side towards the wide section.
Figure 7.10: Temperature Gradient across the Casing and 70% Eccentric Cement Sheath in Instant, Fast, and Slow Heating
Rate Scenarios Respectively
The highest compression damage occurred in the instant heating scenarios for the cement sheath
with 70% eccentricity is also be attributed to the highest thermal flux magnitude detected at the
cement sheath interface with the casing and the formation (Figure 7.11) subjected to the instant
heating scenario.
121 |
ADE | Figure 7.11: Heat Flux Magnitude at the Interface of the Cement Sheaths with the Casing and the Formation Subjected
Instant Heating Scenarios
The narrower sides of the cement sheath with 70% and 50% eccentricity are also experiencing
some disking cracks as a result of steel casing expansion and highly unbalanced stress distribution
in these cases. The thermal expansion coefficient and thermal conductivity of are both higher for
the steel compared to the cement thermal properties which resulted in uneven thermal strains of
the steel casing and the cement sheath led to the creation of disking cracks in this section.
In order to have a more general indicator to compare all the different scenarios, we defined a
global compression damage indicator (D ) as follows.
c
𝐷
=∑𝑁(𝑑𝑐)
(7.22)
𝑐 0
𝑁
where d is the local compression damage magnitudes for all the nodes within the cement sheath,
c
and N is the number of nodes with associated compression damage, i.e. excluding nodes where
the damage is zero.
Figure 7.12 shows the global compression indicator values versus the degree of eccentricity
for heating scenarios.
122 |
ADE | Figure 7.12: Global Compression Damage Indicator vs. Eccentricity for Different Heating
Scenarios under High Temperature Changes
The critical effect of eccentricity and application of controlled heating rates on the integrity of
cement sheaths again can be confirmed by considering the compression damage contour in Figure
7.9, and the global compression damage indicator in Figure 7.12. The global compression damage
indicator reaches the highest value in the case of cement sheath with 70% eccentricity subjected
to instant heating. The magnitude of global compression damage indicator is higher for all the
cement sheaths subjected to instant heating. The magnitude of global compression damage in cases
subjected to controlled heating rates is similar which is indicative of the benefit of finding a
reasonable heating rate to reduce the cement sheath damage.
7.4.2. Tensile Damage Considering Heating Scenarios
The tensile cracking susceptibility is investigated using the tensile damage contours and a global
tensile damage indicator. The state of local tensile damage (cracking) contours within the cement
sheath after mechanical-thermal loading is shown in Figure 7.13. All the damage contours were
scaled from zero to one for comparison purposes. As can be seen in Figure 7.13, the tensile damage
is more localised in comparison with compression damage as shown in Figure 7.9. Considering
the localisation of tensile damage contours demonstrates the important role of eccentricity in the
distribution of tensile stress within the cement sheath again as the tensile damage occurred only
the narrower side of the cement sheaths.
123 |
ADE | Figure 7.14 demonstrates the magnitude of global tensile damage indicators within the cement
sheath subjected to heating scenarios.
Figure 7.14: Global Tensile Damage Indicator vs. Eccentricity during Heating Scenarios
As can be seen in Figure 7.14 the magnitude of global tensile damage reaches the highest value
in the case with 70% eccentricity subjected to the instant heating which emphasises the
conservative aspect of monotonic simulations.
The global tensile damage indicators within the cement sheath with 30% eccentricity subjected
to controlled heating rates are higher than the instant heating scenario in Figure 7.14. In these
scenarios for the 30% eccentric cement sheath with controlled heating rates, the wellbores
gradually warm up which allows the heating flux to transmit entirely within the model which leads
to the higher temperature gradient and consequently creation of higher thermal strains within the
narrower side of the cement sheath. The narrower side of the cement sheath with 30% eccentricity
shows more resistance to the casing expansion in comparisons with the cement sheath with 50%
and 70% eccentricity due to having the highest thickens among them. The high resistance of the
cement sheath with 30% eccentricity leads to high contact shear stresses and subsequently high
tensile damage. Theses observation of tensile cracks indicates that the magnitude of local and
global tensile damage is more dependent on the wellbore geometry rather than the heating rates.
7.5.Influence of Cooling Scenarios along with Pressure Operated on Wellbore-2
Wellbore-2 in the cooling scenarios is subjected to 18 MPa pressure at the inner wall of the
casing along with temperature variations with the initial temperature is assumed to be constant for
all the model components as 573.15 K. Temperature variations of ∆T = 303.15 K−
573.15 K considering three different cooling rates were applied at the inner wall of the casing.
125 |
ADE | In addition, the compression/shear damage was observed at the wider side of the cement sheath
with 70% eccentricity while in heating scenarios the heating scenarios compression damage was
mainly concentrated at the narrower side of the cement sheath. The maximum local compression
damage (d ≈0.3) occurred at the narrower side of the cement sheath with 70% eccentricity
c
subjected to controlled heating rates. The process of cooling down happens gradually in the
controlled heating rates which allow the wellbore to contract further and leads to higher thermal
strains.
Figure 7.16: Global Compression Damage Indicator during Cooling
Figure 7.16 shows the global compression damage indicator computed according to Equation
(7.14). As can be seen in Figure 7.16 the effect of cooling rates on the cement sheath compression
damage is minimal due to the dominant effect of pressurizing the wellbore and in-situ stresses
confinements within the range studied in this paper. These observations are in a good agreement
with the numerical-experimental study performed on the impacts of thermal cycling on wellbore
integrity during CO injections by Roy, Walsh, Morris, Iyer, Hao, Carroll, Gawel, Todorovic and
2
Torsæter [158].
7.5.2. Tensile Damage Considering Cooling Scenarios
The tensile damage occurred in the cooling scenarios was relatively low comparing to the
heating scenarios. The maximum local tensile damage occurred at the narrower side of the cement
0.5oC
sheath with 70% eccentricity (d =0.21) subjected to the slowest cooling rate ( ). The low
t min
tensile damage magnitudes are attributed to the dominant compressive effect of the mechanical
127 |
ADE | Figure 7.19 shows the corresponding thermal strains for the three different cooling rates. As
can be seen in Figure 7.19 the gradient of thermal strain corresponding to the slowest rate is the
steepest which resulted in higher global tensile damage among the three rates. As applying thermal
loads in this scenario with the slowest rate provides more time for the thermal flux to be
transmitted across the wellbore which led to the sharpest gradient of thermal strains.
Figure 7.19: Thermal Strain Gradient for the Selected Path
Considering Different Cooling Rates
7.6. Susceptibility of Forming Micro Annuli
The integrity of the cement sheath bonds in two extreme eccentric cement sheaths studied (30%
and 70%) are examined through a contact stiffness degradation index in this section. Two arbitrary
paths were selected at the cement sheath interfaces as shown in Figure 7.20 along with the starting
locations of the selected paths. Both paths start at the narrowest side and end at the thickest part
of the cement sheaths.
Starting Points
Figure 7.20: Cement Sheath Interfaces with the Casing and Rock Formation
129 |
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