The Influence of Cyclic Loading on the Mechanical Properties of Well Cement

Abstract

The cyclic loading generated by injection and production operations in underground gas storage facilities can lead to fatigue damage to cement sheaths and compromise the integrity of wellbores. To investigate the influence of cyclic loading on the fatigue damage of well cement, uniaxial and triaxial loading tests were conducted at different temperatures, with maximum cyclic loading intensity ranging from 60% to 90% of the ultimate strength. Test results indicate that the compressive strength and elastic modulus of well cement subjected to monotonic loading under high-temperature and high-pressure (HTHP) testing conditions were 14–21% lower than those obtained under ambient testing conditions. The stress–strain curve exhibits stress–strain hysteresis loops during cyclic loading tests, and the plastic deformation capacity is enhanced at HTHP conditions. Notably, a higher intensity of cyclic loading results in more significant plastic strain in oil-well cement, leading to the conversion of more input energy into dissipative energy. Furthermore, the secant modulus of well cement decreased with cycle number, which is especially significant under ambient test conditions with high loading intensity. Within 20 cycles of cyclic loading tests, only the sample tested at a loading intensity of 90% ultimate strength under an ambient environment failed. For samples that remained intact after 20 cycles of cyclic loading, the compressive strength and stress–strain behavior were similar to those obtained before cyclic loading. Only a slight decrease in the elastic modulus is observed in samples cycled with high loading intensity. Overall, oil-well cement has a longer fatigue life when subjected to HTHP testing conditions compared to that tested under ambient conditions. The fatigue life of well cement increases significantly with a decrease in loading intensity and can be predicted based on the plastic strain evolution rate.

1. Introduction

Natural gas, as a widely used energy source, is known for its relatively low carbon emissions (compared to petroleum and coal) and abundant reserves. However, the use of natural gas exhibits seasonal fluctuations, requiring additional storage space during periods of low demand. Among various gas storage methods, depleted oil and gas reservoirs can provide ample storage space, which is also cost-effective and reliable. In fact, they currently account for approximately 80% of the total underground gas storage (UGS) capacity [1,2]. Therefore, depleted oil and gas reservoirs are seen as a promising type of gas storage.

Unlike conventional oil and gas wells, the wellbores used for UGS experience alternating loads generated by gas injection and production operations [3]. Although the stress fluctuations generated by these operations are typically lower than the strength of the cement sheath, well cement is susceptible to fatigue failure under cyclic loading. This ultimately leads to increased well repair costs or even the abandonment of UGS wells [4]. Furthermore, there is a shortage of large-scale and high-quality UGS facilities in China. The primary sources of large gas reservoirs are located in the central and western regions, which are characterized by deeper burial depths and higher temperature and pressure environments [5,6]. These factors can negatively affect the long-term wellbore annulus sealing integrity of UGS facilities [7,8,9,10]. To ensure the safe operation of gas storage facilities, several theoretical studies have been conducted on the mechanical sealing integrity of the wellbore. Chu et al. [11] developed an elastoplastic analysis model based on the Mohr-Coulomb theory to investigate the generation and development of micro-annulus during the loading and unloading of internal pressure in the casing. They found that the generation of wellbore micro-annulus was caused by the interfacial tensile stress exceeding the bond strength. Liu et al. [12] developed an analytical method that used in-situ stress as the initial stress and found that existing evaluation methods overestimated the compressibility of the cement sheath. Additionally, laboratory physical simulation tests have been conducted to investigate the in-situ performance of cement sheaths [13,14,15,16]. Goodwin et al. [13] used steel casing with different internal diameters to simulate the effects of temperature and pressure on the cement sheath by injecting hot oil and applying pressure. They found that the circumferential force of the casing induced by high temperature or internal pressure produced a shearing force at the cement/casing interface, resulting in failure at the cement/casing interface or radial fracturing of the cement sheath. Yuan et al. [14] designed a simplified wellbore to study the radial deformation of the cement sheath by applying cyclic loads. Other simulation tests have also been conducted, suggesting that cyclic loading can lead to radial cracking in the cement sheath when stress reaches its fatigue limit, resulting in sealing failure and loss of integrity [15,16]. However, most of these simulation tests did not consider the cement damage mechanism under high temperatures and confining pressure conditions. As the fatigue failure of cement sheaths is a cumulative process, it is difficult to quantitatively analyze the influence of cyclic loading on the mechanical and deformation characteristics of cement sheaths using large-scale laboratory simulations.

The deformation and fatigue damage of well cement under cyclic loading are inevitably accompanied by crack initiation and propagation, which can be characterized by energy conversion and dissipation. Material deformation and failures are accompanied by changes in energy, which exist in the form of dissipated energy [17]. Therefore, the instability of well cement under cyclic loading is primarily driven by energy. However, research on using energy analysis methods to study material instability has mostly focused on rock materials [18,19,20], with limited research on well cement [21]. Considering the complex internal structure of well cement, especially in high-temperature and high-pressure (HTHP) environments [22], further investigation is warranted.

In this study, monotonic and cyclic loading tests were conducted on well cement under uniaxial and triaxial conditions to investigate the effects of temperature, confining pressure, and intensity of cyclic loading on the mechanical properties and deformation characteristics of well cement before, during, and after cyclic loading tests. Additionally, from an energetic perspective, the conversion of energy and changes in dissipated energy were analyzed. Finally, a fatigue life prediction model for well cement was established to predict fatigue life in different environments.

2. Experimental Method

2.1. Cement Slurry Design

The bulk materials used in this study include Class G oil-well cement, anti-strength-retrogression agent, silica fume, suspending agent, and ductility-enhancement materials. The cement additives used include retarder, defoamer, fluid loss additive, and dispersant. The design of the cement slurry formula is presented in

Table 1. The formula of cement slurry.

Materials	Dosage (%bwoc)
Aksu cement	100
Anti-strength retrogression agent	35
Silica fume	4
Ductility-enhancement agent	4
Suspending agent	1
Dispersant	1
Water	52
Retarder	3
Fluid loss additive	3.5
Defoamer	0.5

. The cement slurry was prepared in accordance with the standard procedure specified in API 10B-2 [23]. The mixture was blended for 15 s at a speed of 4000 rpm using a laboratory mixer, followed by another 35 s of mixing at 12,000 rpm. The mixed cement slurry was then poured into cylindrical stainless-steel molds measuring 25 × 70 mm. Subsequently, the filled molds were placed in a curing autoclave at 140 °C and 20 MPa for 7 days. After reaching the designated curing age, the specimens were removed from the steel molds and cut into a standard size with a diameter of 25 mm and a height of 50 mm. Prior to testing, the specimens were maintained in a saturated state (i.e., wrapped with wet paper towels and sealed in plastic bags).

2.2. Experimental Method

In this study, the Changchun Zhantuo Triaxial Testing Machine (model TAW-1000) was employed to perform both monotonic and cyclic tests under both HTHP and ambient conditions (Figure 1). The equipment has a maximum axial pressure of 1000 KN, a maximum confining pressure of 100 MPa, and a maximum testing temperature of 160 °C. The axial failure stress (σ₁) can be derived from Equation (1), and the elastic modulus (E) and poisson’s ratio can be derived from Equations (2) and (3), respectively.

$${\mathrm{\sigma }}_1={\displaystyle \frac{{\mathrm{F}}_{\max }}{A}}$$

(1)

$$\mathrm{E}={\displaystyle \frac{\Delta {\mathrm{P}}_{(25-40\%)}}{\Delta {\mathrm{\epsilon }}_{\mathrm{a}(25-40\%)}}}$$

(2)

$$\mathrm{u}={\displaystyle \frac{\Delta {\mathrm{\epsilon }}_{\mathrm{l}(25-40\%)}}{\Delta {\mathrm{\epsilon }}_{\mathrm{a}(25-40\%)}}}$$

(3)

where, F_max is the maximum axial failure load of well cement, N; A is the compression area of cylindrical samples, mm²; ΔP_(25–40%), Δε_a(25–40%), and Δε_l(25–40%) are the changes in stress, axial strain, and lateral strain, respectively, corresponding to the linear portion of the stress–strain curve, i.e., approximately 25–40% of ultimate strength.

It should be noted that all triaxial tests were performed under undrained conditions, where pore water was not allowed to flow out. The cyclic loading test program is illustrated in Figure 2, in which the axial load is alternately cycled within a predefined range between the upper and lower load limits. In this study, the lower limit of cyclic loading was set to 10% ultimate strength (0.1 P), while the upper limit was set to 0.9 P, 0.8 P, 0.7 P, and 0.6 P. The ultimate strength is the maximum deviatoric stress on the stress–strain curve (i.e., the failure point) for confined tests. A loading rate of 100 N/s was employed for the investigation. In this study, three specimens were used in the ambient monotonic loading test, and one sample was used for each confining pressure in the high temperature monotonic loading test. In the cyclic loading test, one specimen was used for each test condition. The use of only one sample for a triaxial test is justified by the excellent reproducibility of the test results, as indicated by our previous study [24].

3. Experimental Results

3.1. Monotonic Loading Test Analysis

Under the same effective confining stress, it is well known that increasing testing temperatures can cause reductions in the compressive strengths of rock and well cement [24]. The mechanical properties of well cement subjected to monotonic loading tests are presented in

Table 2. Monotonic loading test results of well cement.

Specimens	Testing Temperature/°C	Testing Pressure/MPa	Compressive Strength/MPa	Elastic Modulus/GPa	Poisson’ Ratio
M1	25	0	80.54	11.70	0.212
M2	25	0	86.16	12.86	0.189
M3	25	0	84.59	12.64	0.188
Average			83.73 ± 2.92	12.4 ± 0.62	0.19 ± 0.014
M4	140	15	67.70	10.96	0.251
M5	140	30	65.45	10.89	0.155
M6	140	45	68.80	11.11	0.178
M7	140	60	67.33	10.25	0.151
M8	140	75	61.77	10.24	0.154
Average			66.21 ± 2.76	10.69 ± 0.41	0.178 ± 0.04

. It is observed that the compressive strength and elastic modulus of well cement obtained under HTHP test conditions decreased by 21% and 14%, respectively, when compared to those obtained under ambient test conditions. The decreases in mechanical properties can be mainly attributed to the high testing temperature, as the test results obtained at various confining pressures are similar. In other words, confining pressure appeared to have very little influence on the mechanical properties of the well cement. This may be due to the fact that all tests were performed under undrained conditions. Similar results have been reported by other studies [25,26,27]. During undrained tests, the pore pressure of saturated well cement will increase with increasing confining pressure, reducing the effective confining stress on the sample. Under confined testing conditions, increasing temperatures could also lead to increases in pore pressure due to the thermal expansion effect. The deformation characteristics of well cement, represented by stress–strain curves, are depicted in Figure 3. It can be observed that the testing environment also affected the stress–strain curve characteristics of well cement. Under the ambient test conditions, the well cement completely lost its ability to resist external loads after stress reached its ultimate compressive strength. However, under the HTHP testing condition, the oil-well cement can still sustain external load after the compressive strength is reached, with residual strengths ranging from 20 MPa to 45 MPa.

3.2. The Mechanical Properties of Well Cement during Cyclic Loading

Previous studies have shown that materials like rocks and cementitious can experience fatigue failure under cyclic loading conditions [28,29]. The stress–strain curves of oil-well cement during cyclic loading are depicted in Figure 4 and Figure 5. It can be seen that oil-well cement tested at ambient conditions experienced fatigue failure within 20 cycles when the upper limit of cyclic loading was set at 0.9 P. However, under HTHP test conditions, all well cement samples completed 20 loading and unloading cycles without failure. The deformation characteristics of well cement were significantly influenced by the intensity of cyclic loading. As the intensity of cyclic loading increased, the hysteresis loops in the stress–strain curves became wider and more dispersed, and the increases in both total strain and cycle number became more prominent. Interestingly, the testing conditions also affected the shape of the hysteresis loops. When subjected to HTHP conditions, the hysteresis loop became larger and took on a diamond shape compared to the oval shape observed under ambient conditions. As hardened oil-well cement contains micropores and micro-cracks as natural defects, its loading and unloading processes generally involve the opening and closing of these defects. While these defects can be readily closed during loading, their opening takes time; hence, strain recovery lags behind stress during unloading, causing hysteresis. The observed differences between HTHP and ambient test conditions can be explained as follows: under triaxial testing conditions (i.e., HTHP conditions), the defects in cement are still compacted by confining pressure during unloading, and their recovery takes even more time, leading to stronger strain recovery lag and bigger stress–strain hysteresis.

The deformation characteristics of the cement sheath can reflect the sealing performance. When significant irreversible deformation occurs, micro-annulus will be formed between cement–casing interfaces or cement-formation interfaces in the wellbore annulus. Figure 6 demonstrates the strain evolution of well cement with increasing loading cycles. Total strain represents the strain value of well cement under the maximum load during each loading cycle, while plastic strain represents the strain value under the minimum load within each cycle. Elastic strain was calculated as the difference between the total strain and the plastic strain. Obviously, both the maximum strain and plastic strain of well cement increased significantly as the upper limit of cyclic loading intensity increased.

Previous research has shown that the primary cause of fatigue failure of cement-based materials was the accumulation of plastic strain with increasing loading cycles [4,30,31]. As shown in Figure 6, when the intensity of cyclic loading was low (e.g., 0.6 P and 0.7 P), the accumulation of plastic strain with cycle number increased extremely slowly under both ambient and HTHP conditions. However, when the intensity of cyclic loading was high (e.g., 0.8 P and 0.9 P), the accumulation of plastic strain increased quickly. At the same loading intensity of 0.9 P, the sample tested under ambient conditions failed with a relatively small increase in plastic strain (total amount of increase was 0.279 from the 1st cycle to failure), whereas the samples tested under HTHP conditions experienced a large increase in plastic strain without failure (total amount of increase was 0.6711 from the 1st cycle to the 20th cycle). Additionally, the initial value of plastic strain (i.e., after the 1st cycle) under the HTHP condition was much greater than that under the ambient condition, which is consistent with the results in [32]. Under the HTHP test condition, oil-well cement was more prone to generating micro-cracks, and hence the cement was more compactable during initial loading. However, under the HTHP test condition, well cement can also sustain a large number of micro-cracks before large fracture surfaces develop (i.e., before complete failure occurs). In contrast, apparent increases in elastic strain with cycle were only observed at high loading intensities (i.e., at 0.8 P and 0.9 P) under the ambient test condition. At all other test conditions, elastic strain remained relatively stable throughout the cyclic testing periods. The initial value of the elastic strain under the HTHP test condition was smaller than that under the ambient condition.

The modulus of well cement is a crucial parameter that characterizes its ability to resist deformation under external loads. Since well cement is not an ideally linear-elastic material, its ability to resist deformation during cyclic loading can be described using the loading and unloading secant modulus [28,31]. The loading secant modulus is defined as the slope of the line connecting the lower limit stress point and the upper limit stress point on the loading curve of each cycle, while the unloading secant modulus corresponds to the slope of the line connecting the upper limit stress point and the lower limit stress point on the unloading curve. Figure 7 illustrates the evolution of the loading and unloading secant moduli of well cement under different testing environments. Due to the presence of initial natural defects in the material, such as micropores and micro-cracks, the loading secant modulus experienced a sharp increase during the first to second loading cycles, reflecting the compaction of these initial defects. Subsequently, under ambient test condition, the loading secant modulus decreased as the number of loading cycles increased and higher cyclic loading intensity resulted in a faster reduction in secant modulus. This suggests that high cyclic loading intensity negatively impacts the internal structure of well cement, causing the formation of additional microscopic defects and ultimately resulting in lower secant moduli. Conversely, under the HTHP test condition, the loading secant modulus remained relatively stable from the second cycle to the twentieth cycle. The unloading secant modulus exhibited similar variation patterns as the loading secant modulus. The unloading secant moduli were higher than the loading secant moduli at ambient test conditions, whereas the unloading secant moduli were approximately equal to the loading secant moduli at HTHP test conditions.

3.3. The Mechanical Properties of Well Cement after Cyclic Loading

After 20 cycles of cyclic loading tests, the well cement samples that had not failed were fully unloaded and further subjected to monotonic loading tests.

Table 3. The mechanical properties of well cement after cycling.

Specimens	Intensity of Cyclic Loading	Testing Temperature/°C	Confining Pressure /MPa	Compressive Strength/MPa	Elastic Modulus/GPa
C2	80%	25	0	77.97	9.89
C3	70%	25	0	83.93	11.31
C4	60%	25	0	81.53	12.67
C5	90%	100	45	69.07	9.83
C6	80%	100	45	67.05	10.56
C7	70%	100	45	66.18	10.96
C8	60%	100	45	62.51	11.69

summarizes the mechanical properties obtained from the monotonic loading test. It should be noted that, due to fatigue failure during the cyclic loading test, sample C1 was not listed in Table 3. After 20 cycles of cyclic loading, the average ultimate compressive strengths of well cement subjected to ambient and HTHP conditions were 81.14 ± 2.45 MPa and 66.20 ± 2.38 MPa, respectively. These values were nearly identical to those obtained without cyclic loading (see Table 2), indicating that the compressive strength of the well cement was hardly affected by the cyclic loading process. Additionally, the elastic modulus of well cement after cycling exhibited a decreasing trend with an increasing intensity of cyclic loading. Specifically, under ambient test conditions, the elastic modulus decreased by 20% with loading intensity increasing from 0.6 P to 0.8 P, while under HTHP test conditions, the elastic modulus decreased by 16% with loading intensity increasing from 0.6 P to 0.9 P. The observed phenomenon could be attributed to the compaction of initial natural defects at low loading intensities (leading to enhanced elastic modulus) and the generation of new microscopic defects at high loading intensities (leading to reduced elastic modulus).

3.4. Energy Analysis during Cyclic Loading

During the cyclic loading and unloading process, well cement undergoes energy input, accumulation, and dissipation. As the well cement was axially compressed during the loading phase, a portion of the input energy was stored as elastic energy within the material, while the remaining energy was converted into dissipative or dissipated energy, which is associated with the closing of existing cracks as well as the generation of new cracks, often causing irreversible deformation. Notably, elastic energy is reversible and completely released during the unloading phase, while dissipative energy is irreversible. Figure 8 shows the energy evolution of well cement under cyclic loading. U represents the total energy input to the well cement from the external environment, which is equal to the area enclosed by the loading stress–strain curve and the horizontal (strain) axis. U_e denotes the reversable elastic energy, corresponding to the area enclosed by the unloading stress–strain curve and the horizontal (strain) axis. U_d represents the dissipated energy, which is the difference between U and U_e. Under the ambient conditions, the three forms of energy were observed to increase slightly with increasing cycle number at high loading intensities but remained relatively stable at low loading intensities. Under the HTHP test condition, there were very few variations in the three forms of energy with the number of cycles, regardless of the cyclic loading intensity.

Figure 9 illustrates the proportion of dissipated energy in total input energy for well cement and its variations with the loading intensity and cycle number under different testing environments. When the intensity of cyclic loading was high (0.9 P), a significant portion of the total energy absorbed by the well cement was converted into dissipated energy. With decreasing cyclic loading intensity, the proportion of dissipated energy in total energy was also reduced. The maximum energy input occurred during the first loading cycle, and the proportion of dissipated energy in total energy was also significantly higher during that cycle. It is obvious that, for all loading intensities, the proportion of dissipated energy in total input energy is significantly higher at the HTHP test condition, compared with that at the ambient test condition.

3.5. Fatigue Failure Model of Well Cement under Alternating Stress

Extensive prior research has established [4,30,32] that the strain evolution of cement-based materials as a function of normalized fatigue cycle count (or loading time) can be categorized into three distinct stages, as illustrated in Figure 10. These stages are: (I) A rapid increase in strain with loading cycles, which can be attributed to the compaction effects of initial natural defects; (II) a linear increase in strain with loading cycles, reflecting the cumulative damage inflicted by fatigue loading on the material; and (III) a rapid increase in strain with loading cycles, indicative of unstable crack propagation within the material prior to failure. During the second stage, the rate of strain evolution with normalized cycle count (or time) remained largely unaffected by loading intensity and loading rate [4,30,32]. Consequently, the rate of strain change within the second stage can be utilized to predict the fatigue life (N_F) of well cement, as demonstrated in Equation (1).

$$\ln {\stackrel{·}{\epsilon }}_{\mathrm{Ⅱ}}=\ln f+\ln {\displaystyle \frac{{\epsilon }_{\mathrm{Ⅱ}a}-{\epsilon }_{\mathrm{Ⅱ}b}}{a-b}}-\ln N_F$$

(4)

where, ${\dot{\epsilon }}_{II}$ represents the rate of strain change with time during the second stage of strain evolution; f denotes the loading frequency (Hz); N_F represents the fatigue life, which is the maximum number of cycles the test sample can sustain before failure; a and b correspond to the normalized loading cycle’s initial (N_a/N_F) and final (N_b/N_F) points within the second stage, respectively; ${\epsilon }_{\mathrm{I}\mathrm{I}a}$ and ${\epsilon }_{\mathrm{I}\mathrm{I}b}$ represent the well cement strain values at points a and b. These strain values can be expressed as total strain or plastic strain, provided that they correspond to the strain value used in ${\dot{\epsilon }}_{\mathrm{I}\mathrm{I}}$. It should be noted that the division of the three different stages may not be obvious with short fatigue lives, especially when the fatigue life is less than 10 cycles, but the rate of strain evolution during the middle stage can still be employed to represent the strain rate evolution during the second stage. Utilizing the relationship between loading frequency and loading period, the rate of strain evolution with loading cycles can be correlated with the rate of strain evolution with time. Assume ${\displaystyle \frac{\partial {\epsilon }_{\mathrm{I}\mathrm{I}}}{\partial n}}$ is the rate of strain evolution with cycle number during the second stage of strain evolution. By combining Equations (4) and (5), Equation (6) can be obtained:

$${\displaystyle \frac{\partial {\epsilon }_{\mathrm{Ⅱ}}}{\partial n}}={\displaystyle \frac{\partial {\epsilon }_{\mathrm{Ⅱ}}}{\partial (t\times f)}}={\displaystyle \frac{1}{f}}\times {\displaystyle \frac{\partial {\epsilon }_{\mathrm{Ⅱ}}}{\partial t}}={\displaystyle \frac{1}{f}}\times {\stackrel{·}{\epsilon }}_{\mathrm{Ⅱ}}$$

(5)

$$\ln {\displaystyle \frac{\partial {\epsilon }_{\mathrm{Ⅱ}}}{\partial n}}=\ln {\displaystyle \frac{{\epsilon }_{\mathrm{Ⅱ}a}-{\epsilon }_{\mathrm{Ⅱ}b}}{a-b}}-\ln N_F$$

(6)

Therefore, the rate of well cement strain evolution with the loading cycle in the second stage can be employed to predict its fatigue life. Note that the key parameter ${\displaystyle \frac{{\epsilon }_{\mathrm{I}\mathrm{I}a}-{\epsilon }_{\mathrm{I}\mathrm{I}b}}{a-b}}$ remains constant regardless of the intensity of cyclic loading and can be obtained by performing a cyclic test on a standard sample until failure. In this study, samples M1 and M5 were selected as the standard samples to calculate such key parameters for ambient and HTHP conditions, respectively. It should be noted that, because sample M5 did not fail during the cyclic loading, its test results can only be used to estimate the conservative values of fatigue life (i.e., the real fatigue life should be higher than the estimated values).

Table 4. Fatigue parameters of well cement under ambient condition (25 °C/0 MPa).

Samples	C1	C2	C3	C4
Maximum loading intensity	90%	80%	70%	60%
Loading rate/(N/s)	100	100	100	100
Loading frequency f/Hz	0.00159	0.00181	0.00209	0.00251
Loading period T/s	629	553	478	398
${\displaystyle \frac{{\epsilon }_{\mathrm{I}\mathrm{I}a}-{\epsilon }_{\mathrm{I}\mathrm{I}b}}{a-b}}$	0.149	0.149	0.149	0.149
${\displaystyle \frac{\partial {\epsilon }_{\mathrm{I}\mathrm{I}}}{\partial n}}$	0.00196	0.00535	0.00215	0.00102
${\dot{\epsilon }}_{\mathrm{I}\mathrm{I}}={\displaystyle \frac{\partial {\epsilon }_{\mathrm{I}\mathrm{I}}}{\partial t}}/$10−5·s−1	3.12	0.967	0.450	0.256
Fatigue life NF	7.5	27 *	68 *	-

* Predicted fatigue life.

and

Table 5. Fatigue parameters of well cement under HTHP condition (140 °C/45 MPa).

Samples	C5	C6	C7	C8
Maximum loading intensity	90%	80%	70%	60%
Loading rate/(N/s)	100	100	100	100
Loading frequency (f/Hz)	0.00193	0.00219	0.00257	0.00308
Loading period (T/s)	519	458	389	324
${\displaystyle \frac{{\epsilon }_{\mathrm{I}\mathrm{I}a}-{\epsilon }_{\mathrm{I}\mathrm{I}b}}{a-b}}$	0.302	0.302	0.302	0.302
${\displaystyle \frac{\partial {\epsilon }_{\mathrm{I}\mathrm{I}}}{\partial n}}$	0.0151	0.00540	0.00297	-
${\dot{\epsilon }}_{\mathrm{I}\mathrm{I}}={\displaystyle \frac{\partial {\epsilon }_{\mathrm{I}\mathrm{I}}}{\partial t}}/$10−5 s−1	2.890	1.179	0.763	-
Fatigue life/NF	>20	>56 *	>102 *	-

* Predicted fatigue life.

show the fatigue strength predicted by the mathematical model. It can be seen that the fatigue life of well cement under HTHP testing conditions was much higher than that under ambient testing conditions. When subjected to a cyclic loading intensity of 0.6 P, well cement exhibited minimal fatigue damage with a strain evolution rate close to zero, suggesting that 60% of ultimate strength is close to the fatigue limit, below which no fatigue failure can occur.

4. Conclusions

(1)

The mechanical properties of well cement are significantly influenced by the testing environment. Specifically, increasing the testing temperature from 25 °C to 140 °C reduces the compressive strength and elastic modulus of well cement by 21% and 14%, respectively. However, the confining pressure has almost no effect on the mechanical properties or stress–strain curves of well cement during undrained tests at high temperatures, possibly due to increased pore pressure with increasing confining pressure.

(2)

The deformation and mechanical properties of well cement are influenced by the cyclic loading intensity and cycle number. Under HTHP conditions, well cement exhibits more pronounced strain hysteresis and greater plastic strain, which leads to a higher conversion rate of input energy into dissipated energy. The secant modulus of well cement decreases with increasing loading intensity at all test conditions and also decreases with increasing cycle number, especially under ambient testing conditions with high loading intensity.

(3)

Samples that can sustain 20 cycles of cyclic loading without failure exhibit similar ultimate compressive strength as the pristine sample not subjected to cyclic loading. However, the elastic modulus of cyclically loaded samples may decrease when the cyclic loading intensity is high.

(4)

Well cement has a longer fatigue life when subjected to HTHP testing conditions compared to that tested under ambient conditions. The fatigue life of well cement increases significantly with a decrease in loading intensity and can be predicted based on the plastic strain evolution rate. At 0.6 P loading intensity (60% of ultimate strength), well cement hardly experiences any fatigue damage.