Strength Behaviors and Constitutive Model of Gas-Saturated Methane Hydrate-Bearing Sediment in Gas-Rich Phase Environment

Abstract

Natural gas hydrates occupy an important position in the development of clean energy around the world in the 21st century. It is of great significance to research the mechanical properties of methane hydrate-bearing sediment (MHBS). In this paper, gas-saturated MHBS were synthesized based on the self-developed triaxial compressor apparatus. The triaxial shear tests were performed at temperatures of 2 °C, 3 °C, and 5 °C and confining pressures of 7.5 MPa, 10 MPa, and 15 MPa. Results indicate that the axial strain process can be divided into three stages: initial elastic deformation, initial yield deformation, and strain softening. When confining pressure is increased, the shear strength of MHBS increases at a lower confining pressure. In contrast, shear strength appears to decrease with increasing confining pressure at a higher confining pressure. There is a negative correlation between temperature and shear strength of MHBS. The initial yield strain of MHBS increases in condition due to the increase in confining pressure and the decrease in temperature. The change in strength degradation is kept within 2 MPa. Using test data, the Duncan-Chang model was modified to describe the strength behaviors of gas-saturated MHBS. The accuracy of the model was verified by comparing calculated values with test data.

1. Introduction

Natural gas hydrates, as an efficient and environmentally friendly energy source, play an important role in the development of clean energy in the world in the 21st century [1]. Natural gas hydrates are crystalline molecular compounds that form under conditions of high pressure and low temperature. A molecule of natural gas hydrate is formed by gas (CH₄, CO₂, etc.) molecules and water molecules (H₂O) [2]. The appearance of natural gas hydrates is similar to ice [3]. The reserves of methane gas occurrence in methane hydrate-bearing sediment (MHBS) in the South China Sea are estimated to be 6.5 × 10¹³ m³ [4]. On the other hand, there are certain risks in the exploitation of MHBS [5]. Due to the shallow buried depth and weak consolidation of MHBS in the sea area, a series of engineering and geological problems, such as wellbore instability, cementing failure, sand production, seabed settlement, and landslides, are prone to occur during drilling and production [6,7]. The instability of MHBS caused by the decomposition of hydrate during exploitation may lead to the collapse of the submarine slope at the continental margin and even cause geological disasters such as submarine landslides, earthquakes, and tsunamis [8]. At the same time, the use of CO₂ replacement for the exploitation of natural gas hydrate has attracted extensive attention from scholars. CO₂ replacement exploitation can not only recover CH₄, but also bury CO₂. It is expected that gas hydrate extraction, carbon burial, and reservoir stability will be achieved simultaneously [9]. Thus, research into the strength behaviors of MHBS is critical for the safe and efficient exploitation of gas hydrate resources.

Natural gas hydrates exist in hydrate reservoirs in the form of pore filling, load bearing, and cementation [10], as shown in Figure 1. When the permeability of natural gas in the reservoir is very low, the permeable gas can completely synthesize the hydrate. There is no free gas in the sediment, which is defined as a water-rich phase environment [11]. On the other hand, if the sediment structure is a fault zone, the permeability of natural gas is high. Free gas exists in the sediment, which is defined as a gas-rich phase environment [12]. In a water-rich phase environment, natural gas is dissolved in water. Gas-dissolved water forms hydrate crystals in sediment pores at low temperature and high pressure. Hydrate crystals float in pores, which is called pore filling. Hydrate crystals do not transfer load or cement the soil skeleton. Their contribution to the sediment strength is frictional rather than cohesive. [13]. With the increase in permeability of natural gas, hydrate crystals contact the soil skeleton and play a role in load transmission and cementation called load bearing [14]. In a gas-rich phase environment, the sediment is unsaturated, and water is distributed at the contact surface of soil particles. Hydrate crystals are synthesized at the gas–water interface and cemented with adjacent soil particles to form the bearing skeleton called cementation [15]. Compared with pore filling and load-bearing, the cementation of hydrates has a more obvious influence on the mechanical properties of sediment [16]. MHBS are more likely to exhibit strain softening and dilatancy [17,18]. Winters et al. elaborated that higher ventilation volume during MHBS synthesis in the laboratory caused methane gas hydrate to play a role in the cementation [19]. Thus, gas-saturated MHBS in a gas-rich phase environment can be synthesized using the circulating gas method to research the mechanical properties of MHBS.

On the basis of the original soil strength test apparatuses, a low-temperature refrigeration chamber and a high-pressure gas inlet and outlet system were added to realize in situ MHBS synthesis [20]. A series of experiments revealed that the mechanical strength of synthetic MHBS was similar to that of natural MHBS [16,21,22,23]. Hyodo et al. reported that higher hydrate saturation, lower temperature, and larger pore pressure increased the peak strength of MHBS [24]. Yun et al. used tetrahydrofuran hydrate to carry out a series of triaxial compression tests. Their research reflected that the mechanical properties of tetrahydrofuran hydrate sediment mainly depended on the skeleton particles when the hydrate saturation was low. When the hydrate saturation was high, the hydrate cemented between skeleton particles [25]. The larger the particle size of the sediment, the lower the pore pressure. Hydrate exists in sediment with a large skeleton particle size [26]. Song et al. confirmed that an increase in confining pressure and strain rate, and a decrease of temperature led to the enhancement of shear strength [27]. Miyazaki et al. found that the cohesion of MHBS was related to hydrate saturation, while the internal friction angle was almost constant [28]. MHBS samples experienced volume shrinkage during the early stages of axial loading and then began to expand [29]. Priest et al. illustrated that MHBS exhibited significant post-peak strain softening. The existence of hydrate significantly reduced the permeability of sediment [30]. Based on a large number of experiments, some scholars establish mechanical models of MHBS. The Duncan-Chang model, which accurately describes the nonlinear elastic relationship of stress–strain, has been widely used for MHBS. Miyazaki et al. introduced parameters such as hydrate saturation and an effective confining pressure to modify the Duncan-Chang model [31]. Subsequently, a lot of scholars corrected the Duncan-Chang model at different conditions [32,33,34].

Aiming at the above-mentioned problems, gas-saturated MHBS were synthesized using the self-developed triaxial compressor apparatus in this paper. The triaxial shear tests were performed at temperatures of 2 °C, 3 °C, and 5 °C and confining pressures of 7.5 MPa, 10 MPa, and 15 MPa. The Duncan-Chang model was modified by using the test data to describe the strength behaviors of MHBS. The accuracy of the model was verified. The research results will give some suggestions on the replacement of CO₂ for hydrate exploitation stably.

2. Materials and Methods

2.1. Triaxial Testing Apparatus

The self-developed temperature-controlled high-pressure triaxial compressor is used in the test to simulate the in situ pressure, temperature, and stress conditions of methane hydrate reservoirs in the deep seabed, as shown in Figure 2 and Figure 3 [35]. The volume of the cell is Φ50 mm × 100 mm. The testing apparatus consists of an axial loading system, a sediment sample synthesis system, a constant temperature control system, and a control measurement system. Axial loading system can provide an axial load range of 0–250 kN. Temperature can be conducted in a range from −35 °C to 50 °C. The thermostatic bath is used to adjust the temperature of the sample with an accuracy of ±0.1 °C by circulating glycol solution as frozen liquid. The constricting pressure can be adjusted from 0 to 30 MPa. The amount of deionized water mixed with sediment sand determines hydrate saturation. Before synthesizing sediment, the saturation is calculated to deduce the volume of deionized water. It is assumed that all deionized water participates in the hydrate formation reaction, and then the saturation is controlled. In the process of the triaxial compression test, mechanical parameters such as temperature, confining pressure, and strain change can be collected and saved in real-time by a computer. The fastest speed of computer data collection can reach six times per second.

2.2. Sample Preparation

The sand of the MHBS skeleton used in the test was drilled from the natural gas-hydrate reservoir on the seabed of the Shenhu Sea area in the South China Sea. Sand grains are able to hinder and decelerate the dissociation process [36]. From Zhang’s study, hydrate is mainly formed in coarser sand [37]. Thus, the sand drilled from the Shenhu Sea was screened by a standard inspection sieve (60–100 meshes). The corresponding particle size was 0.25–0.38 mm. The median particle size was 0.1122 mm, and the porosity was 35%, as shown in Figure 4. Methane and nitrogen were used in the test, with a purity of 99.99%. Deionized water was made in the laboratory.

Methane hydrate was synthesized in a high-pressure reactor using the circulating gas method. The dried sand was mixed with quantitative deionized water, stirred, and sealed for 24 h. The sand was partially water saturated. A rubber sleeve was placed on the pedestal and fixed with a rubber band. The permeable plate, metal filter screen, and filter paper were installed on the pedestal in turn. The outside of the rubber sleeve was fixed with a metal jacket. Using the layered filling method, the wet sand was added into the rubber sleeve and compacted with a compactor every few spoons. The sample was divided into 10 compacted layers. Based on this method, it was assumed that the sample was homogeneous. The filter paper, metal filter screen, and permeable plate were placed on top of the sample, and a two-way intake cover was finally installed. After the sample was filled, confining pressure was applied. Meanwhile, pore pressure raised synchronously, but it was always 1 MPa lower than confining pressure. The final confining pressure was 7 MPa, and the temperature was controlled at 20 °C for 12 h. The temperature of the sample was reduced to 2 °C to form methane hydrate, which lasted for 72 h. After the sample was synthesized, low-temperature methane gas was injected in a circular pattern to ensure the sample was gas saturated.

3. Results

Undrained shear was adopted in the test. Previous research showed that the submarine temperature is between 1.45 and 9.00 °C in the Shenhu Sea [38]. Considering that a lower temperature is more conducive to the synthesis of MHBS, the temperatures during shearing were set to 2 °C, 3 °C, and 5 °C. MHBS in the Shenhu Sea buried at a depth of about 125–200 m below the seabed [39]. Therefore, the shear confining pressures were 7.5 MPa, 10 Mpa, and 15 Mpa. The test finished when the axial strain reached 15%.

3.1. Experimental Reproducibility Verification

When the tests were completed, all the specimens indicated a plastic deformation. A shear dilation was observed during the tests and the medium of the specimens protruded in a lateral direction, as shown in Figure 5.

Reproducible experiments are conducted at confining pressures of 7.5 MPa and temperatures of 5 °C. Cases 1–3 use MHBS as the experimental material, while Cases 4–5 use wet pure sand. The data from the stress–strain curves is then analyzed and displayed in Figure 6. Whether it is MHBS or pure sand, the difference in curves is very small. It is demonstrated that the test apparatus possesses good reproducibility. Furthermore, the curves of MHBS can be seen to be higher than those of pure sand. This phenomenon suggests that the hydrate will significantly increase the strength of MHBS.

3.2. Stress–Strain Curves

Figure 7 shows the stress–strain curves of MHBS at various temperatures and confining pressures. It can be seen that the curves exhibit the shape of a hyperbola. The stress–strain behavior indicates that the axial strain process can be divided into three stages: initial elastic deformation (OA), initial yield deformation (AB), and strain softening (after B). As the axial strain increases, the deviator stress of the MHBS increases in an approximately straight line until it reaches 1–2% of the axial strain. MHBS exhibit elastic deformation during initial elastic deformation. Then, the deviator stress increases slowly, and the growth rate gradually decreases until reaching a peak point (Point B), and MHBS presents as both elastic and plastic deformation. With a further increase of axial strain (after B), the deviator stress decreases and gradually approaches a constant value. MHBS manifests as a strain-softening behavior, and the shear dilation can be observed. According to the literature, strain softening is defined as a gradual loss of shear strength with an axial strain after a peak strength has been reached [40]. The reason for the strain-softening phenomenon is that plastic deformation accumulates continuously and the cementation of the methane-hydrate and soil particles is destroyed gradually [32].

3.3. Shear Strength

The maximum value of deviator stress is selected as shear strength $q_f$. Figure 8 shows the shear strength of MHBS at different confining pressures at temperatures of 2 °C, 3 °C, and 5 °C respectively. It can be observed that the temperature and confining pressure affect the shear strength of MHBS distinctly. When the confining pressure is less than 10 MPa, the increase in confining pressure leads to an increase in the shear strength of MHBS. However, when the confining pressure exceeds 10 Mpa, shear strength of MHBS appears to decrease to some degree with the increase in confining pressure. In addition, the influence of temperature on the trend of shear strength is roughly similar. Shear strength increases as the temperature decreases.

According to the literature [41], an increase in confining pressure can not only limit the deformation of MHBS, but also induce damage of MHBS. With confining pressure increasing, relative sliding, rotation, and overturning of adjacent particles occur between soil particles at a lower confining pressure, which need to consume more energy to deform MHBS. MHBS is in a compacted state at this stage and promotes cementation between skeleton particles. This status contributes to the cohesion of MHBS expansion. Shear strength manifests an augmentation. On the other hand, a higher confining pressure can induce the destruction of MHBS. At higher confining pressures, a part of the particle breaks, overcomes the biting force, and crosses adjacent particles [42]. This leads to a reduction in shear strength. Furthermore, an excessive confining pressure causes a local pressure-melt effect of ice and methane hydrate crystals, resulting in an MHBS cohesion decline. Shear strength manifests a decrease [43].

3.4. Initial Yield Strain

Figure 9 shows the confining pressure-dependence curves of initial yield strength at different temperatures. The critical point A, where the slope has an obvious change on the stress–strain curves marks the end of elastic deformation and the beginning of yield deformation. The deviator stress at point A is defined as the initial yield strength $q_0$ [44]. It can be observed that $q_0$ increases with the increase in confining pressure at a constant temperature. When the confining pressure is invariant, $q_0$ increases with the decline in temperature. The expression of $q_0$ for confining pressure ${\sigma }_3$ and temperature $T$ are calculated by fitting as shown in Equation (1).

$$q_0=\left(0.06{\sigma }_3-0.69\right)T^2+\left(3.36-0.52{\sigma }_3\right)T+2{\sigma }_3-3.08$$

(1)

Figure 10 shows the relationship between initial yield strength $q_0$ and shear strength $q_f$ at various temperatures and confining pressures. There is a unique relationship between initial yield strength and shear strength. This relationship is not dependent upon temperature and confining pressure, which is similar to frozen silt. Initial yield strength can be inferred from shear strength, independent of temperature and confining pressure. Equation (2) expresses it as a linear equation.

$$q_0=0.55q_f=0.96$$

(2)

3.5. Strength Degradation

With the attainment of shear stress, the stress–strain curves descend with the increase in axial strain, which is defined as strain softening. Deviator stress corresponding to the axial strain of 15% is selected as degraded strength $q_d$. Strength degradation $\delta {\sigma }_3=q_0-q_d$ represents the maximum degradation possible [45]. The strength degradation of MHBS is computed and tabulated in

Table 1. Strength degradation of MHBS.

Temperature	2 °C			3 °C			5 °C
$\mathbf{Confining}\mathbf{Pressure}{\mathit{\sigma }}_3,\mathbf{MPa}$	7.5	10	15	7.5	10	15	7.5	10	15
Shear strength $q_0$, MPa	16.22	24.69	20.22	13.47	18.73	17.20	4.34	10.35	6.51
Degraded strength $q_d$, MPa	14.88	24.04	18.12	11.83	18.25	16.19	4.22	9.63	5.86
Strength degradation $\delta {\sigma }_3$, MPa	1.34	0.65	2.10	1.64	0.48	1.01	0.12	0.72	0.65

. The variation of strength degradation of MHBS changes slightly but generally stays within 2 MPa. However, the strength degradation of MHBS at 5 °C is obviously bigger. This is due to the lower temperature, which leads to a stronger cementation of hydrate. With the axial stress increasing continuously, the hydrate gets broken, and the particles of sediment migrate and rearrange, thus the strength degradation becomes larger.

4. Constitutive Model

The Duncan-Chang model [46] is applied to analyze the triaxial test of MHBS described above. According to the nonlinear elastic Duncan-Chang model, the corresponding expression for the hyperbolic stress–strain function is given as Equation (3).

$$\frac{{\epsilon }_a}{q}=a+b{\epsilon }_a$$

(3)

where $q={\sigma }_1-{\sigma }_3$ is deviant stress; ${\epsilon }_a$ is axial strain; $a$ is the inverses of the initial tangent elastic modulus $E_i$ as Equation (4); $b$ is the inverses of the asymptotic value of the deviator stress $q_{ult}$ as Equation (5).

$$a=\frac{1}{E_i}$$

(4)

$$b=\frac{1}{q_{ult}}$$

(5)

Thus, $a$, $b$, $E_i$, and $q_{ult}$ can be determined by the linear approximation of the experimental $\left({\epsilon }_a/q\right)~{\epsilon }_a$ data. $a$ is the intercept on the vertical axis of $\left({\epsilon }_a/q\right)~{\epsilon }_a$. $b$ is the slope of the line of $\left({\epsilon }_a/q\right)~{\epsilon }_a$. The triaxial test data of MHBS was processed in accordance with the relationship of $\left({\epsilon }_a/q\right)~{\epsilon }_a$, which is indicated in Figure 11. The values of $a$ and $b$ are computed and tabulated in

Table 2. Values of the coefficients of $a$ and $b$.

Temperature, °C	2 °C		3 °C		5 °C
Coefficients	$\mathit{a}$	$\mathit{b}$	$\mathit{a}$	$\mathit{b}$	$\mathit{a}$	$\mathit{b}$
7.5 MPa	8.78 × 10−4	5.56 × 10−2	1.47 × 10−3	5.14 × 10−2	1.76 × 10−3	2.21 × 10−1
10 MPa	5.55 × 10−4	3.56 × 10−2	6.41 × 10−4	4.76 × 10−2	9.25 × 10−4	9.05 × 10−2
15 MPa	3.30 × 10−4	4.98 × 10−2	4.02 × 10−4	5.65 × 10−2	5.75 × 10−4	1.59 × 10−1

.

Figure 12 shows the variation of the initial elastic modulus $E_i$ with temperature $T$ and confining pressure ${\sigma }_3$. The initial elastic modulus can be increased by increasing the confining pressure and decreasing the temperature.

Based on the Duncan-Chang model, the relationship between the initial elastic modulus and the confining pressure can be expressed as Equation (6).

$$E_i=K{P}_a{\left(\frac{{\sigma }_3}{P_a}\right)}^n$$

(6)

where $P_a$ is atmospheric pressure; $K$ and $n$ are test parameters. With further transformation, Equation (6) can be rewritten as Equation (7).

$$\lg \frac{E_i}{P_a}=\lg K+n\lg \frac{{\sigma }_3}{P_a}$$

(7)

It can be seen that $\lg \left(E_i/P_a\right)$ and $\lg \left({\sigma }_3/P_a\right)$ have a linear relationship. $\lg K$ is the intercept on the vertical axis, and $n$ is the slope of the line of $\lg \left(E_i/P_a\right)~\lg \left({\sigma }_3/P_a\right)$. The values of $\lg \left(E_i/P_a\right)$ and $\lg \left({\sigma }_3/P_a\right)$ are computed and tabulated in

Table 3. Values of the coefficients of $\lg \left(E_i/P_a\right)$ and $\lg \left({\sigma }_3/P_a\right)$.

Temperature, °C		2 °C	3 °C	5 °C
Coefficients	$\mathbf{lg}\left({\mathit{\sigma }}_3/{\mathit{P}}_{\mathit{a}}\right)$	$\mathbf{lg}\left({\mathit{E}}_{\mathit{i}}/{\mathit{P}}_{\mathit{a}}\right)$	$\mathbf{lg}\left({\mathit{E}}_{\mathit{i}}/{\mathit{P}}_{\mathit{a}}\right)$	$\mathbf{lg}\left({\mathit{E}}_{\mathit{i}}/{\mathit{P}}_{\mathit{a}}\right)$
7.5 MPa	1.88	4.06	3.83	3.75
10 MPa	2.00	4.26	4.19	4.03
15 MPa	2.18	4.48	4.40	4.24

, and the linear fitting relationship is shown in Figure 13.

In addition, the Duncan-Chang model only considers the effect of the confining pressure on the initial elastic modulus. Parameter $T$ is introduced into the Duncan-Chang model as Equations (8) and (9).

$$\lg K=0.39T^2-2.95T+5.77$$

(8)

$$n=-0.18T^2+1.35T-0.56$$

(9)

From Equations (6)–(9), the initial elastic modulus $E_i$ as shown in Figure 14 can be rewritten as Equation (10).

$$E_i={10}^{0.39T^2-2.95T+5.77}{P}_a{\left(\frac{{\sigma }_3}{P_a}\right)}^{-0.18T^2+1.35T-0.56}$$

(10)

The damage ratio $R_f=q_f/q_{ult}$ can be transformed through Equation (5) as $R_f=b{q}_f$, and the calculated values of $R_f$ are shown in Figure 15. The dependence of $R_f$ on confining pressure and temperature is not clear. $R_f$ holds for a variety of soils ranging from 0.7 to 1. For simplicity, $R_f$ is set as a constant independent of confining pressure and temperature as $R_f=0.92$.

It is convenient to use the instantaneous elastic modulus $E_t$ in the stress increment analysis as Equation (11).

$$E_t=\frac{\partial q}{\partial {\sigma }_3}=E_i{\left(1-R_f\frac{q}{q_f}\right)}^2$$

(11)

The stress–strain curves of MHBS are fitted by the above constitutive model. Figure 16 presents the model predictions as well as the experimental results. It can be seen that the calculated curves match the experimental curves well. However, the Duncan-Chang model can only predict the strain hardening curve. There are some errors in the calculation of the strain softening phenomenon, which needs more attention in further studies.

5. Conclusions

In this study, a series of triaxial compression tests of synthetic gas-saturated MHBS were performed at various high confining pressures and temperatures. The modified Duncan-Chang constitutive model was employed to describe the stress–strain curves of MHBS based on the experimental data. The following are the main conclusions:

The stress–strain behavior indicates that the axial strain process can be divided into three stages: initial elastic deformation, initial yield deformation, and strain softening. The deviator stress increases as an approximately straight line. Then, the deviator stress increases slowly, and the growth rate decreases gradually until reaching a peak point. With a further increase in axial strain, the deviator stress decreases and gradually approaches a constant value.

When the confining pressure is less than 10 MPa, the increase in confining pressure leads to the enlargement of the shear strength of MHBS. However, when the confining pressure exceeds 10 MPa, the shear strength appears to decrease to some degree with the increase in confining pressure. Shear strength increases as temperature decreases. The initial yield strain of methane hydrate increases with the increase in confining pressure and the decrease in temperature. The variation in strength degradation changes very little.

Using the Duncan-Chang constitutive model, modifications are made to describe the mechanical behavior of gas-saturated MHBS. A comparative analysis of experimental and simulation results indicates the calculated curves match the experimental curves well. Furthermore, some corrections aimed at the calculation of the strain softening phenomenon will continue in future research.