Abstract
Mechanical properties of methane hydratebearing-sediments (MHBS) are basic parameters for safety analysis of hydrate exploration and exploitation. Young's modulus, cohesion, and internal friction angle of hydratebearing sediments synthesized in laboratory, are investigated using tri-axial tests. Stress-strain curves and strength parameters are obtained and discussed for different compositions and different hydrate saturation, followed by empirical expressions related to the cohesion, internal friction angle, and modulus of MHBS. Almost all tested MHBS samples exhibit plastic failure. With the increase of total saturation of ice and methane hydrate (MH), the specimens' internal friction angle decreases while the cohesion increases.

gas hydrate is distributed extensively in ocean sediments, permafrost and deep lakes [1] [2] [3] [4] .

In recent years, methane hydrate has attracted great interest in scientific communities for the following reasons [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] :

(1) It is conservatively estimated that more than 50% of the 18.8 terratonnes of organic carbon present on the earth is in the form of gas hydrate, found either in marine sediments or in permafrost. Extraction of methane from hydrates could provide a future energy resource [6] .

(2) Methane is a greenhouse gas 20 times potent than carbon dioxide [7] . The volume of methane currently bound in hydrate is thought to be many thousand times that held in the atmosphere. Either the changes in temperature or pressure can cause the instability of methane hydrate-bearingsediments (MHBS). Loss of stability in seafloor hydrate could lead to sea-level rising and global climate change [8] [9] [10] .

(3) Evidence suggests that dissociation of methane hydrate (MH) can be a trigger for long run-out submarine landslides [11] [12] [13] [14] [15] [16] . It is reported that the reason of the biggest Storegga slide was hydrate dissociation induced by the increase of seawater temperature [14] . These huge events are known to have led in the past to major tsunamis and widespread flooding and devastation along continental littoral [15] .

(4) Oil and gas exploration are now extending far off the continental shelf where hydrates can be present in relatively shallow layer below the sea bed. There is a concern that hydrocarbon exploration and development activities may trigger hydrate dissociation which may result in seabed slope instability [17] [18] [19] . Recently, the deepwater horizon explosion in gulf may be caused by dissociation of MH since the drilling rig had reached the sediments where the pressure and temperature is proper for hydrate formation [20, 21] .

A common concern with submarine MH is to identify its global and local occurrence, concentration and form, and the most suitable methods for exploitation. Because MH exists only under very restricted conditions, it is difficult to determine their presence and properties by drilling or to bring undisturbed specimens to the laboratory for testing [22] . Another method of locating and characterizing hydrates comes from marine seismic geophysical records. However, the development, validation, and optimization of seismic surveying techniques requires an understanding of the relationship between sediment type, hydrate form, content, and physical and mechanical properties of the sediment [23, 24] . Properties of the porous host sediment affect the morphology and extent of hydrate growth, which in turn may alter the host sediment properties [25, 26] .

The difficulty of drilling, sampling and synthesizing sediment containing MH is responsible for a lack of related mechanical properties [25] . Winters et al. [27] studied the mechanical properties of MHBS. The test results showed that both the strength and the compressive wave speed of MHBS are greater than that for sediments without methane hydrate (MH). The increment depends on the saturation and distribution of MH, and the properties of sediments.

Hyodo [28] investigated the relationship between mechanical properties and temperature, pore pressure, confining pressure and MH saturation. Experiments of Masui et al. [29, 30] showed that the strength of in-situ samples is the same as that of synthetic samples, although the stress-strain curves are different. The differences come mainly from the differences in initial void ratio and grain sizes. These results indicated that synthetic samples can be used to model in-situ MHBS.

Wei et al. [31] tested deformation and failure properties of carbon dioxide hydrate in sands, and the results showed that shear modulus and strength increased with carbon dioxide hydrate saturation. Yun et al. [32] used the tetrahydrofuran hydrate bearing sediments to conduct series tests, with hydrate saturation, confining pressure, and particle size of soil taken into account. Miyazaki et al. [33] studied artificial methane hydrate bearing sediments using drained tri-axial tests and found that the strength and stiffness increased with hydrate saturation and effective confining pressure.

At present, the mechanical data of hydrate-bearing sediments are lack quantitatively and the mechanical properties are expressed qualitatively. In this paper, tri-axial tests are carried out at low temperature (below freezing temperature) and high pressure conditions to study mechanical properties (e.g., elastic modulus and strength) of MHBS with various MH saturations. For comparison, the mechanical parameters of frozen ice-bearing sediments and water saturated sand are also obtained by tri-axial tests. Empirical formulas for strength parameters of MHBS are presented to explore the relationship between contents and strength parameters of hydrate, ice and soil skeleton.
Equipment and methods

Test apparatus
Tests were carried out in an apparatus developed at the Institute of Mechanics, Chinese Academy of Sciences (Fig. 1) . The apparatus provides confining pressures ranging from 0 to 14 MPa with an accuracy of 0.5% (Fig. 2) and temperatures from -20
Test apparatus
• C to 20
Test apparatus
• C with an accuracy of 2.5% (Fig. 3) . The maximum back-pressure provided by a gassupply cylinder is 10 MPa. This allowed the necessary high pressure and low temperature conditions for achieving the formation of MH. A gas flow-meter was used to measure the total gas percolating into the specimen which is initially partially water saturated. Then hydrate is formed under proper pressure and temperature (Fig. 4 [34] ). A gas collection system with a volume of 6 L was designed to measure the gas released from the specimen during dissociation (Fig. 5) . In this system the water in two cylinders is pushed out once the methane gas is released from the specimen. The process can be described as follows: When methane gas is released from MHBS, the gas pressure is larger than the pressure in the collecting system which is equal to atmosphere pressure, so the water is pushed out until the pressures at two sides are equal. The gas volume is then obtained by measuring the displaced water with a precision of about 0.5%, because the gas volume under pressure of 0.1 MPa and temperature of 0
Test apparatus
• C is almost equal to the displaced water (the solubility of methane in water under this condition is ignorable).
Gas collector Graduated cylinder
Fig. 5 Gas collection system
Test procedure
The material tested was a fine silty sand with dry density of 1.6 g/cm 3 , maximum void ratio of 0.949 and minimum void ratio of 0.454. The grain size distribution is shown in Fig. 6 . Phase equilibrium curves of MH in porous sediments are quite different from that of bulk MH. As the pore size decreases, higher pressure or lower temperature are needed for phase equilibrium compared with that of bulk hydrates. For an example, the super-cooling degree can be over −10
Test procedure
• C as the pore size decreases to less than 10 nm. A radius of 60 nm is regarded as the critical value for the temperature offset in hydrate formation [35] .
Test procedure
Three thin sections were cut from the top, middle, and bottom of a specimen by the sample cutting method in soil mechanics. Then the sections were covered with thin water layers on the surface and observed under a microscope with an accuracy of 2 µm to obtain pore-throat characteristics when the boundaries of pores and throats could be distinguished. Shown in Fig. 7 are the observed pore-throat structures. The channel between two neighboring white lines is called the pore throat; the area crossed by two or more throats is called the pore. For the specimen under test the mean radius of pore-throats is over 50 µm, thus the offset of temperature and pressure can be neglected. In lack of microscopic observation technology, we took several specimens out of the cell after MH formation, checked whether the specimens were intact and hard. We found that the samples required great physical force to break the synthesized intactness.
Test procedure
The procedure can be described in five steps as follows:
Test procedure
(1) Prepared sand specimens (initially 10% water saturated in quality) with diameter and height of 39.1 mm and 80 mm, respectively, in a copper mold. Each specimen was sealed within a butyl rubber membrane (0.1 mm in thickness) to prevent subsequent methane gas migration.
Test procedure
(2) Placed the specimen into the tri-axial cell and pushed water into the specimen until a desired saturation was reached. A thermo-couple was attached to the cell wall to measure the temperature. The cell top was then fitted and a confining pressure was slowly applied. As this methane gas was pressurized into the specimen, the flow rate was controlled by a gas flow-meter. The rubber membrane was expanded to allow a peripheral gas supply around the specimen at a proper pressure difference between confining pressure and gas pressure, which was favorable for more uniform MH specimen. Then a pore pressure of 2.5 MPa was maintained 24 hours to induce methane gas dissolution in the pore water.
Test procedure
(This step was not necessary for preparing frozen specimens without MH.) (3) The specimen temperature was gradually lowered to -9
Test procedure
• C into the hydrate-stability range to form MH. When the gas flow into the specimen became insignificant and the pore pressure was stable after MH was formed, the temperature was increased to a value slightly above the hydrate-stability range for at least three hours to trigger MH dissociation. Temperature was decreased again to form MH. This process was repeated at least twice to increase the saturation of MH (It is called temperature oscillation method [36] ).
Test procedure
(4) After the specimen were prepared, the excess gas was drained. An axial load was applied to the top of the specimen by a ram at a constant rate of 0.9 mm/min until the strain reached 15% during which the tests were un-drained. The axial displacement, axial force and pore pressure of the specimen were measured and recorded (here for un-drained condition, the volume strain could be neglected).
Test procedure
(5) The temperature was increased to dissociate the MH in the specimen. The volume of methane released during dissociation was measured by the collection system described in Sect. 2.1.
Test results and analysis
Hydrate saturation greatly affects the mechanical properties of MHBS. The initial water saturation of the specimen was controlled by the volume of water, V w , percolated into the specimen before the formation of hydrate. The volume of methane gas, V g , consumed during the formation of gas hydrate was obtained using the collecting system shown in Fig. 5 . The hydrate saturation S H and the ice saturation S I can be calculated from Eqs. (1)-(3) .
Test results and analysis
The hydrate saturation is obtained by the gas volume measured
Test results and analysis
The ice saturation is equal to the difference between the total water content and that contained in the hydrate
Test results and analysis
where
Test results and analysis
V p is the pore volume of the sediments, p 0 is 0.1 MPa, T 0 is 273.15 K, R is gas constant with a value of 8.31 J/(mol · K), M h and M w are molar mass of hydrate and water respectively, ρ h and ρ w are density of hydrate and water respectively, N h is hydrate number (the number of water molecules corresponding to one methane molecule). In this paper MH was formed at temperatures below 0
Test results and analysis
• C, thus the pores were filled by ice except for MH. The Mohr-Coulomb type criterion is adopted here to describe the strength of MHBS [37] 
Test results and analysis
Here τ is the yield stress, σ is the normal stress, θ is the internal friction angle, C is the cohesion, ε H is the volume percentage of hydrate, ε I is the volume percentage of ice, ε r is the volume percentage of soil skeleton. ε H , ε I , and ε r are defined as function of porosity φ, S H and S I Volume percentage of MH:
Test results and analysis
Volume percentage of ice:
Test results and analysis
Volume percentage of soil skeleton: ε r = 1 − φ.
Test results and analysis
We note that: (1) The specimen temperature was set at -9
Test results and analysis
• C because MH specimens were synthesized relatively easily and the experimental results could be properly analyzed by solid mechanics; (2) When the total water saturation was above 90%, the sample would collapse or become curved during water saturating or MH formation due to partial soil softening; (3) The total saturation was set at about 80%, the respective saturation of ice and methane hydrate was changed above 10%, and then the change of mechanical parameters can be observed and analyzed clearly.
Test results for water-saturated sand
To obtain mechanical properties of non hydrate-bearing sand specimens, three water saturated sand specimens (named A1, B1 and C1) were prepared and tested in consolidated drained conditions under confining pressures of 2.5 MPa, 5 MPa and 10 MPa, respectively. Figure 8a shows the stress-strain curves of the saturated sand. It is shown that the failure modes are all plastic (the stress increases linearly with the strain initially, and keeps constant until the engineering failure strain of 15% is achieved). The deviatoric stress (σ 1 − σ 3 , σ 1 is the axial stress, σ 3 is the confining stress.) reaches about 20 MPa under a confining pressure of 10 MPa. Figure 8b shows that the effective cohesion and internal friction angle of the sand skeleton are 0.1 MPa and 30.5
Test results for water-saturated sand
• , respectively. 
Test results for frozen sediment
Three specimens of frozen sediment (named A2, B2 and C2) with ice saturations of 86.3%, 83.2% and 85.5% were prepared for triaxial tests under confining pressures of 2.5 MPa, 5.0 MPa and 10.0 MPa, respectively. Failure modes of the three specimens are all plastic (Fig. 9a) . The deviatoric stress reaches 10 MPa and 6 MPa under confined pressure of 10 MPa and 2.5 MPa, respectively. The drained conditions in tests of water saturated samples and the expansion in ice samples could lead to larger deviatoric stresses relative to that of frozen sediment. The cohesion and internal friction angle are 2.8 MPa and 9.6
Test results for frozen sediment
• , respectively (Fig. 9b) . Compared with the water saturated sand, the cohesion increases significantly while the internal friction angle decreases significantly. 
Test results for MHBS
In this section, three groups of tests (each group had three specimens) were carried out to analyze the effects of MH saturation under undrained conditions. The three specimens in the first group (named A3, B3, and C3) were prepared with MH saturations of 29.6%, 31.1% and 31.1%, respectively, and ice saturations of 59.6%, 55.5% and 51.1%, respectively. Triaxial tests were carried out under confining pressures of 2.5 MPa, 5.0 MPa and 10.0 MPa, respectively. Because the pores were filled with ice and hydrate, there was no pore pressure in tests. Figure 10a shows the stress-strain curves from the three tests. This group of specimens had low MH saturations and high ice saturations, which may be the reason that the failure modes are all plastic similar to those of the iced sediment shown in Fig. 6a . The deviatoric stresses are 6 MPa and 3 MPa under confined pressure of 10 MPa and 2.5 MPa, respectively. The results indicate that the strength of MHBS were lower than those of frozen sediments, which is in agreement with the results of Winters et al [27] . Figure 10b shows that the cohesion and internal friction angle of MHBS are 1.2 MPa and 8.5
Test results for MHBS
• , respectively. Note that the total saturations of ice and MH in the three specimens are 89.2%, 86.6% and 82.2%, respectively. It is clear that the cementation of ice with grains is greater than that of MH compared with the test results for frozen sediment. The internal friction of MHBS is affected weakly by the formation of either ice or MH in the pores. The three specimens in the second group (named A4, B4, and C4) had MH saturations of 43%, 48.9% and 51.3% and ice saturations of 31.5%, 26.4% and 24.4%, respectively. The same confining pressures of 2.5 MPa (A4), 5 MPa (B4) and 10 MPa (C4) as those in the tests for the first group were used. Figure 11a shows the stress-strain curves of the three specimens. It can be seen that B4 and C4 are still plastic failure. However, A4 presents a brittle failure (there is a peak stress). The cohesion and internal friction angle are 1.16 MPa and 12.4
Test results for MHBS
• , respectively (Fig. 11b) . Although the cohesion is almost the same, the internal friction angle is greater than that of A3, B3 and C3. The three specimens in the third group (named A5, B5 and C5) with MH saturations of 26.3%, 26.3% and 30%, respectively, and ice saturations of 45.8%, 45.8% and 42.6%, respectively, were also carried out under confining pressures of 2.5 MPa, 5 MPa and 10 MPa, respectively.
Test results for MHBS
The measured stress-strain curves, as can be seen from Fig. 12a , are similar to that of A4, B4, and C4. However, this group of specimens has a smaller cohesion and a greater internal friction angle, which are 0.7 MPa and 19.3
Test results for MHBS
• (Fig. 12b) , respectively, compared with those for the first and second groups.
Test results for MHBS
Generally speaking, with the increase of total saturation of ice and MH, the specimens' internal friction angle decreases while the cohesion increases. Under the same saturation, the increase of ice content leads to the increase of cohesion. It is believed that the cementation (The grains are cemented by ice and MH.) is enhanced with the increase of total saturation of ice and hydrate, which leads to strong cementing force among sand particles. The strength and modulus of sediments containing MH are influenced by a number of factors: strain rate, temperature, consolidation stress, grain size, density and even gas hydrate cage occupancy. The pores of MHBS are filled with MH, which increases the tendency for dilation during shear [28] . Although the water saturated sand used in tests is not cemented, cementation develops once it contains MH.
A model for the strength of MHBS

An empirical formula for strength parameters of MHBS under different conditions
It is well known that the formation of MH will change the properties of MHBS. Generally, ten parameters determine the mechanical properties: ε H , ε r , ε I , C H , C r , C I , T e , P e , ε, θ H , θ r and θ I , in which C H , C r , C I are cohesions of hydrate, soil skeleton, and ice, respectively, T e , P e are phase equilibrium temperature and pressure of hydrate formation, ε is shear rate, θ H , θ r , θ I are friction angles of hydrate, soil skeleton, and ice. Hyodo [28] showed that the temperature and pressure during the formation of gas hydrate sediments (GHS) had no effects on the strength. The strain rate (strain in unit time) in tests is generally small. So assuming that T e , P e ,ε have little effect on MHBS's modulus, meanwhile the cohesion of matrix is C r = 0, thus the cohesion and the internal friction angle of MHBS are functions of ε H /C H , ε I /C I and ε H /tan θ H , ε I /tan θ I , ε r /tan θ r , respectively
An empirical formula for strength parameters of MHBS under different conditions
Let ε t = ε H + ε I and assume
An empirical formula for strength parameters of MHBS under different conditions
and
An empirical formula for strength parameters of MHBS under different conditions
Here, the unknowns C H , C I , tan θ H , tan θ I and tan θ r are expected to be obtained by fitting experimental data of ε t , ε H , ε I , ε r , C and tan θ.
An empirical formula for strength parameters of MHBS under different conditions
Comparison of the test data for the cohesion and internal friction is presented in Table 1 and the data obtained from Eqs. (1) and (2) are shown in Tables 2 and 3 . It is shown that the test data are in good agreement with the fitted values. 
An empirical formula for elastic modulus of MHBS
A dimensionless expression is expressed as follows using the ten parameters ε H , ε r , ε I , E H , E r , E I which are important in determining the elastic modulus of MHBS
An empirical formula for elastic modulus of MHBS
Since ice, hydrate and the sand skeleton of MHBS are all solid phases, a linear expression of modulus related to the Reuss model and weighted average of the three-phase Wood equation [38, 39] is developed
An empirical formula for elastic modulus of MHBS
in which a, b and c are three coefficients. According to the test data shown in Table 4 , E H and E I are valued as 8 200 MPa and 9 500 MPa, respectively [40] , the coefficients a, b and c may be obtained by the least squares method: a = 117.30, b = 34.55, c = 1.03. The comparison of modulus by tests and theory are shown in Table 5. It is shown that they are quite close to each other. 
Conclusions
Consolidated undrained tri-axial tests on frozen, methane hydrate-bearing sediments and consolidated drained triaxial tests on water saturated sand were carried out in a special tri-axial apparatus at the Institute of Mechanics, Chinese Academy of Sciences. The confining pressures adopted were 2.5 MPa, 5 MPa and 10 MPa. It is shown that the formation of ice and MH significantly change the mechanic properties of MHBS. An empirical model for the strength of MHBS is presented. Failure modes of samples are generally plastic (e.g. there is no peak stress). In tests presented in this paper, the maximum saturation of hydrate is less than 40%, thus the MHBS is still stress dependent [41] . With the increase of total saturation of ice and MH, the specimens' internal friction angle decreases a little while the cohesion increases. Under the same saturation, the increase of ice content leads to an increase of cohesion.
Conclusions
The modulus and internal friction angle and cohesion of MHBS can be expressed as a linear combination of each component's saturation and corresponding parameters (e.g., modulus, friction and cohesion) based on mixed theory.
Conclusions
The results in this paper are expected to serve as a reference for estimating Yang's modulus and strengths of MHBS from the contents of hydrate, ice and skeleton. In the future, more widely varied saturations, the homogeneity of hydrate distribution in sediments and the effects of compositions and microstructures of soil skeleton will be considered. Further, the formulation of plastic deformation of hydratebearing sediments referred to the model by Aryanpour and Farzaneh [42] will be explored also which can provide a basis for evaluating the possible geo-hazards in hydrate exploitation.

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