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Article

A Nonlinear Elastic Model for Triaxial Compressive Properties of Artificial Methane-Hydrate-Bearing Sediment Samples

1
Institute for Geo-Resources and Environment, National Institute of Advanced Industrial Science and Technology, 1-1-1 Higashi, Tsukuba, Ibaraki 305-8567 Japan
2
Methane Hydrate Research Center, National Institute of Advanced Industrial Science and Technology, 16-1 Onogawa, Tsukuba, Ibaraki 305-8569 Japan
3
Department of Environmental Science, Toho University, 2-2-1 Miyama, Funabashi, Chiba 274-8510 Japan
*
Author to whom correspondence should be addressed.
Energies 2012, 5(10), 4057-4075; https://doi.org/10.3390/en5104057
Submission received: 28 June 2012 / Revised: 1 October 2012 / Accepted: 16 October 2012 / Published: 19 October 2012

Abstract

:
A constitutive model for marine sediments containing natural gas hydrate is essential for the simulation of the geomechanical response to gas extraction from a gas-hydrate reservoir. In this study, the triaxial compressive properties of artificial methane-hydrate-bearing sediment samples reported in an earlier work were analyzed to examine the applicability of a nonlinear elastic constitutive model based on the Duncan-Chang model. The presented model considered the dependences of the mechanical properties on methane hydrate saturation and effective confining pressure. Some parameters were decided depending on the type of sand forming a specimen. The behaviors of lateral strain versus axial strain were also formulated as a function of effective confining pressure. The constitutive model presented in this study will provide a basis for an elastic analysis of the geomechanical behaviors of the gas-hydrate reservoir in the future study, although it is currently available to a limited extent.

1. Introduction

Natural gas hydrates are anticipated to be a promising energy resource [1,2,3,4]. The mechanical property of a gas hydrate reservoir is considered to be essential for sustainable production, because it will affect the stability of a wellbore or other subsea structures, the occurrence of geohazards and gas productivity [5,6,7,8].
A constitutive model for gas-hydrate-bearing sediments is required for the simulation of the geomechanical response to gas extraction from a gas-hydrate reservoir. Laboratory studies are useful for constructing a constitutive model for gas-hydrate-bearing sediments. However, few constitutive model for gas-hydrate-bearing sediments has been published [9], although there have been reports on laboratory studies concerning the triaxial compressive properties of natural and artificial gas-hydrate-bearing sediment samples [10,11,12,13,14,15,16,17,18,19].
An elastic constitutive model may be practically useful for the preliminary simulation of the mechanical behavior of a reservoir on the condition that we use it with particular attention to the applicable range. In this study, the application of the Duncan-Chang model, a nonlinear elastic model, to the analysis of the triaxial compressive properties of artificial methane-hydrate-bearing sediment samples reported mainly in an earlier work [19] was examined. The dependences of the mechanical properties on methane hydrate saturation and effective confining pressure were considered.

2. Review of Triaxial Compression Test

The methods and results of triaxial compression test for artificial methane-hydrate-bearing sediments were described in our previous studies [12,14,16,17,18,19]. In this section, the test methods and results are briefly described.

2.1. Test Methods

Host specimens were prepared by freezing a cylindrical unsaturated sand specimen. The skeleton of each specimen was formed by Toyoura sand (average particle size D50 = 230 × 10−6 m, fine fraction content Fc = 0.0%), No. 7 silica sand (D50 = 205 × 10−6 m, Fc = 1.1%) or No. 8 silica sand (D50 = 130 × 10−6 m, Fc = 11.5%). These three types of sand contain SiO2 as a major component. The relative density of each host specimen was over 96%. The average porosities of the host specimens formed by Toyoura sand, No. 7 silica sand and No. 8 silica sand were 37.8%, 38.6% and 41.2%, respectively. The relative density of each host specimen was over 96%. Each host specimen was 50 mm in diameter and 100 mm in length.
The artificial methane-hydrate-bearing sediment samples, which are hereafter referred to as a “hydrate-sand specimen,” were prepared from the frozen host sand by the following procedure. First, a cell pressure of 0.5 MPa, 1 MPa, 2 MPa or 3 MPa was applied. Second, the pore air in the specimen was replaced by the methane at 268 K or less. Third, the pore gas pressure was increased to 8 MPa at a rate of approximately 0.7 MPa/min, whereas the cell pressure was increased to 8.5 MPa, 9 MPa, 10 MPa or 11 MPa at the same rate. Forth, the temperature inside the triaxial cell was increased to 278 K. The cell pressure, pore gas pressure and temperature were then kept constant for 24 h during the hydrate formation period. The pore gas pressure was controlled by a pressure-reducing valve attached to the methane gas cylinder. Finally, water was injected into the specimen to replace the residual gaseous methane in the pores of the specimen. We also prepared “sand specimens,” i.e., water-saturated specimens of densely packed sand sediment without hydrate by omitting the methane hydrate formation process from the above-described procedure.
Axial loading was conducted at an axial strain rate of 0.1%/min under a drained condition while maintaining the pore pressure at 8 MPa, the cell pressure at 8.5 MPa, 9 MPa, 10 MPa or 11 MPa and the temperature at 278 K. The methane hydrate existing in the specimen was dissociated by reducing pore pressure after the axial loading. Then, the volume of released methane was measured using a gas flow meter. Assuming that all the released gas have been converted to the hydrate during the axial loading, the initial volume of the methane hydrate formed in the specimen, and thus the methane hydrate saturation Sh, was calculated from the volume of released methane, where Sh is the initial volume percentage of methane hydrate in the pores of the specimen. The methane hydrate saturation Sh of each hydrate-sand specimen used in this study was below 60%.

2.2. Test Results

Figure 1 shows the deviator stress q plotted against the axial strain εa and lateral strain εl for the sand specimens (a: Sh = 0%) and hydrate-sand specimens (b: Sh = 27%–34%, c: Sh = 41%–45%) formed by Toyoura sand under the effective confining pressures σ3’ of 0.5 MPa, 1 MPa, 2 MPa and 3 MPa. In Figure 1, as well as in Figure 2, Figure 3, Figure 4 and Figure 5, solid lines represent the experimental results reported in the earlier work [19] and broken lines represent the calculated results with the constitutive model which will be described later. Axial/lateral strain was calculated by dividing the axial/lateral displacement measured with the LVDTs using the initial height/diameter of the specimen in this paper. Note that the lateral displacement was measured at the middle height of the specimen. A positive strain denotes compression in this paper. The q increases and the slopes of the q-εa and q-εl curves decrease until q reaches a peak, and then q gradually decreases. Under a given σ3’, the larger the Sh, the larger the strength (maximum deviator stress) qf and the initial tangent elastic modulus Ei. For a given Sh, the specimen has a larger qf and larger Ei and becomes more ductile under a higher σ3’, as in the case of other geological materials.
Figure 2 shows the εl plotted against the εa for the sand specimens (a: Sh = 0%) and hydrate-sand specimens (b: Sh = 27%–34%, c: Sh = 41%–45%) formed by Toyoura sand under the σ3’ of 0.5 MPa, 1 MPa, 2 MPa and 3 MPa. All the εl-εa curves are concave upward in this graph. A specimen of higher Sh has a tendency to expand further in the lateral direction. It is also found that εl tends to decrease with increasing σ3’ at a given εa.
Figure 1. Experimental (solid lines) and calculated (broken lines) deviator stress q versus axial strain εa and lateral strain εl for specimens formed by Toyoura sand with methane hydrate saturation Sh of (a) 0%; (b) 27%–34% and (c) 41%–45%. Experimental results are taken from the reference [19].
Figure 1. Experimental (solid lines) and calculated (broken lines) deviator stress q versus axial strain εa and lateral strain εl for specimens formed by Toyoura sand with methane hydrate saturation Sh of (a) 0%; (b) 27%–34% and (c) 41%–45%. Experimental results are taken from the reference [19].
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Figure 2. Experimental (solid lines) and calculated (broken lines) lateral strain εl versus axial strain εa for specimens formed by Toyoura sand with methane hydrate saturation Sh of (a) 0%; (b) 27%–34% and (c) 41%–45%. Experimental results are taken from the reference [19].
Figure 2. Experimental (solid lines) and calculated (broken lines) lateral strain εl versus axial strain εa for specimens formed by Toyoura sand with methane hydrate saturation Sh of (a) 0%; (b) 27%–34% and (c) 41%–45%. Experimental results are taken from the reference [19].
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Figure 3 and Figure 4 show the q-εa and q-εl curves and the εl-εa curves, respectively, for the specimens formed by No. 7 silica sand. Figure 5 and Figure 6 show the q-εa and q-εl curves and the εl-εa curves, respectively, for the specimens formed by No. 8 silica sand. It is found that the relationships among the q, εa and εl for the specimens formed by No. 7 and No. 8 silica sand are qualitatively similar to those for the specimens formed by Toyoura sand.
Figure 3. Experimental (solid lines) and calculated (broken lines) deviator stress q versus axial strain εa and lateral strain εl for specimens formed by No. 7 silica sand with methane hydrate saturation Sh of (a) 0%; (b) 23%–28% and (c) 49%. Experimental results are taken from the reference [19].
Figure 3. Experimental (solid lines) and calculated (broken lines) deviator stress q versus axial strain εa and lateral strain εl for specimens formed by No. 7 silica sand with methane hydrate saturation Sh of (a) 0%; (b) 23%–28% and (c) 49%. Experimental results are taken from the reference [19].
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Figure 4. Experimental (solid lines) and calculated (broken lines) lateral strain εl versus axial strain εa for specimens formed by No. 7 silica sand with methane hydrate saturation Sh of (a) 0%; (b) 23%–28% and (c) 49%. Experimental results are taken from the reference [19].
Figure 4. Experimental (solid lines) and calculated (broken lines) lateral strain εl versus axial strain εa for specimens formed by No. 7 silica sand with methane hydrate saturation Sh of (a) 0%; (b) 23%–28% and (c) 49%. Experimental results are taken from the reference [19].
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Figure 5. Experimental (solid lines) and calculated (broken lines) deviator stress q versus axial strain εa and lateral strain εl for specimens formed by No. 8 silica sand with methane hydrate saturation Sh of (a) 0%; (b) 19%–23% and (c) 44%–50%. Experimental results are taken from the reference [19].
Figure 5. Experimental (solid lines) and calculated (broken lines) deviator stress q versus axial strain εa and lateral strain εl for specimens formed by No. 8 silica sand with methane hydrate saturation Sh of (a) 0%; (b) 19%–23% and (c) 44%–50%. Experimental results are taken from the reference [19].
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Figure 6. Experimental (solid lines) and calculated (broken lines) lateral strain εl versus axial strain εa for specimens formed by No. 8 silica sand with methane hydrate saturation Sh of (a) 0%; (b) 19%–23% and (c) 44%–50%. Experimental results are taken from the reference [19].
Figure 6. Experimental (solid lines) and calculated (broken lines) lateral strain εl versus axial strain εa for specimens formed by No. 8 silica sand with methane hydrate saturation Sh of (a) 0%; (b) 19%–23% and (c) 44%–50%. Experimental results are taken from the reference [19].
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Figure 7 shows the strength qf plotted against the methane hydrate saturation Sh for the specimens formed by Toyoura sand, No. 7 silica sand and No. 8 silica sand as the skeleton. The strength qf increases with Sh under each effective confining pressure σ3’. It can be seen that there is little difference in strength among the three types of sand and hydrate-sand specimens formed by Toyoura sand, No. 7 silica sand and No. 8 silica sand at the same Sh and σ3’.
Figure 7. Strength qf versus methane hydrate saturation Sh. Experimental results are taken from the reference [19].
Figure 7. Strength qf versus methane hydrate saturation Sh. Experimental results are taken from the reference [19].
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3. Nonlinear Elastic Constitutive Model

3.1. Duncan-Chang Model

Miyazaki et al. [20] suggested a possibility of application of Duncan-Chang model to the stress-strain relationships of hydrate-sand specimen formed by Toyoura sand. Duncan and Chang [21] described the following nonlinear elastic constitutive model:
q = ε a a + b ε a
where a and b are coefficients related to the initial tangent elastic modulus Ei and the asymptotic value of the deviator stress qult, as shown by:
a = 1 E i
b = 1 q u l t
Equation (1) can be rewritten as:
ε a q = a + b ε a
Thus, a and b, and ultimately Ei and qult, can be determined by the linear approximation of the experimental (εa/q)-εa data. In this study, the (εa/q)-εa data at the stress level (q/qf) in the range of 70% to 95% were used to determine a and b, as recommended in Duncan and Chang’s study [21].

3.2. Strength

The four curves shown in Figure 7 represent the calculated strengths for the effective confining pressures σ3’ of 0.5 MPa, 1 MPa, 2 MPa and 3 MPa with the following equations as functions of σ3’ and Sh:
q f = 2 cos φ 1 sin φ c 0 + α S h β + 2 sin φ 1 sin φ σ 3 '
c0 = 0.30 MPa
φ = 33.8°
α = 4.64 × 10−3
β = 1.58
The form of Equation (5) is determined on the basis of the Mohr-Coulomb failure theory and the observation by Masui et al. [14] indicating that the cohesion increases with Sh and the internal friction angle φ is almost independent of Sh. The values of the parameters in Equation (5) were determined as Equations (6–9) using all of the experimental data shown in Figure 7.

3.3. Initial Tangent Elastic Modulus

Figure 8 shows Ei, or the inverse of a, plotted against the methane hydrate saturation Sh. It is found that Ei shows an increasing trend with Sh and σ3’, although quite a variation can be seen. Janbu [22] has shown that the relationship between Ei and σ3’ may be expressed as:
E i = K p a ( σ 3 ' p a ) n
where pa = atmospheric pressure, K = a modulus number and n = the exponent determining the rate of variation of Ei with σ3’, indicating that Ei is proportional to the n-th power of σ3’. Thus, in this study, it is assumed that Ei can be expressed as a function of Sh and σ3’ as:
Ei(Sh, σ3’) = ei(Sh) ∙ σ3n
where ei(Sh) = the initial tangent elastic modulus as a function of Sh under σ3’ of 1 MPa, or Ei(Sh, 1). When Sh = 0%, the n and ei in Equation (11) for sand specimens formed by Toyoura sand, No. 7 silica sand and No. 8 silica sand are respectively estimated by the least-squares regression:
n = { 0.608 ( for Toyoura sand ) 0.466 ( for No . 7   silica sand ) 0.356 ( for No . 8   silica sand )
e i 0 = { 398 MPa ( for Toyoura sand ) 344 MPa ( for No . 7   silica sand ) 241 MPa ( for No . 8   silica sand )
where ei0 = ei for a sand specimen (Sh = 0%).
Figure 8. Initial tangent modulus Ei versus methane hydrate saturation Sh for specimens formed by (a) Toyoura sand; (b) No.7 silica sand and (c) No.8 silica sand. The curves are calculated using Equation (11).
Figure 8. Initial tangent modulus Ei versus methane hydrate saturation Sh for specimens formed by (a) Toyoura sand; (b) No.7 silica sand and (c) No.8 silica sand. The curves are calculated using Equation (11).
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Figure 9 shows the normalized initial tangent elastic modulus Ei*(Sh), or the initial tangent modulus divided by (ei0σ3n), plotted against Sh for the hydrate-sand specimens:
E i * ( S h ) = E i ( S h , σ 3 ' ) e i 0 σ 3 ' n = e i ( S h ) σ 3 ' n e i 0 σ 3 ' n = e i ( S h ) e i 0
Equation (14) is derived from Equation (11) and indicates that the relationship between Ei* and Sh is independent of σ3’. Because ei0 is equal to ei for a sand specimen, Ei* should be 1 at Sh = 0%. In this study, it is assumed that the increase of Ei* is proportional to Sh to the δ-th power:
E i * ( S h ) = e i ( S h ) e i 0 = 1 + γ S h δ
where γ and δ are estimated by the least-squares regression for the Ei*−Sh shown in Figure 9 as:
γ = 3.90 × 10−4
δ = 2.10
Because a wide variation is seen in Figure 9, there are various possible expressions for the Ei*–Sh relationship other than Equation (15).
Figure 9. Normalized initial tangent elastic modulus Ei* versus methane hydrate saturation Sh for hydrate-sand specimens. The curve is calculated using Equation (15).
Figure 9. Normalized initial tangent elastic modulus Ei* versus methane hydrate saturation Sh for hydrate-sand specimens. The curve is calculated using Equation (15).
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The parameter a in Equation (1), or the inverse of Ei, is given by Equations (11–17). The initial tangent modulus Ei calculated using Equations (11–17) is shown in Figure 8.

3.4. Failure Ratio

Duncan and Chang [21] defined the failure ratio Rf as:
R f = q f q u l t
Strength qf can be obtained from each test result as shown in Figure 7. Asymptotic value of the deviator stress qult, which is equal to the reciprocal of b as expressed by Equation (3), also can be obtained from each test result, because b can be determined by the linear approximation of the experimental (εa/q)-εa data as seen in Equation (4). Thus Rf, which is the ratio of the strength qf to qult, is obtained from each experimental data. Figure 10 shows Rf plotted against Sh. Because the dependence of Rf on σ3’ is not clear as well as a number of different soils, we assume that Rf is independent of σ3’. Rf shows a weak decrease with increasing Sh. However, for simplicity, we assume that Rf is independent of Sh:
Rf = 0.82
which is the average Rf for the sand and hydrate-sand specimens. The data presented in Figure 10 is insufficient to determine the function form expressing the reduction in Rf to Sh. The data for Sh of higher than 60% is necessary to formulate Rf more accurately.
Figure 10. Failure ratio Rf versus methane hydrate saturation Sh.
Figure 10. Failure ratio Rf versus methane hydrate saturation Sh.
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Eventually, the Duncan-Chang model (Equation (1)) may be applicable to the determination of the q-εa relationship of the hydrate-sand specimens using Equations (2–3, 5–9 and 11–19).
It can be conveniently used in incremental stress analysis, because the tangent elastic modulus Et corresponding to any point on the q-εa curve is derived in the simple form:
E t = q ε a = ( 1 R f q q f ) 2 E i

3.5. Lateral Strain

In this section, we examine the following expression for the relationship between the lateral strain εl and the axial strain εa:
εl = −fεa2gεa (f > 0, g > 0)
where f and g are fitting parameters. The tangent Poisson’s ratio υt and the initial tangent Poisson’s ratio υi are expressed by:
ν t = ε l ε a = 2 f ε a + g
ν i = ε l ε a | ε a = 0 = g
Fitting prameters f and g can be adjusted using each experimental data on εl-εa by the least-squares regression. Figure 11 shows υi plotted against Sh. The Sh dependence of υi is weak, while the σ3’ dependence of υi is considerably apparent. Thus, we assume that υi is independent of Sh. Kulhawy and Duncan [23] described that υi generally decreases with increasing σ3’ in the form:
ν i = G F log ( σ 3 ' p a )
where G = υi at 1 atm and F = the rate of change of υi with σ3’. We decided to apply a similar concept to Equation (24) to the dependence of υi on σ3’ as follows:
υi (σ3’) = υi1m ∙ log(σ3’)
where υi1 = the initial tangent Poisson’s ratio υi under σ3’ of 1 MPa; and m = the coefficient determining the rate of decrease in υi with increasing log(σ3’). The least-square logarithmic regression of experimental data gives the values of υi1 and m for the sand and hydrate-sand specimens:
ν i 1 = { 0.394 ( for Toyoura sand ) 0.324 ( for No . 7   silica sand ) 0.253 ( for No . 8   silica sand )
m = { 0.253 ( for Toyoura sand ) 0.245 ( for No . 7   silica sand ) 0.257 ( for No . 8   silica sand )
Figure 12 shows f plotted against υi. Although some variation can be seen in Figure 12, f appears to be approximately proportional to υi as:
f = l × υ i
where l = the proportional constant corresponding to the slope of the line shown in Figure 12:
l = 0.186
Using Equation (28), Equations (21) and (22) can be rewritten as:
εl (σ3’) = −(lεa2 + εa) ∙ υi (σ3’)
υt (σ3’) = (2 ∙ lεa + 1) ∙ υi (σ3’)

3.6. Calculated Deviator Stress versus Axial and Lateral Strain

The calculated q-εa, q-εl and εl-εa curves for the sand specimens (a: Sh = 0%) and hydrate-sand specimens (b: Sh = 30%, c: Sh = 43%) under effective confining pressures σ3’ of 0.5 MPa, 1 MPa, 2 MPa and 3 MPa using Equations (1) and (21) are shown in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6. It is found that the experimental q-εa, q-εl and εl-εa curves are reasonably reproduced by Equations (1) and (21).
Figure 11. Initial tangent Poisson’s ratio υi versus methane hydrate saturation Sh for specimens formed by (a) Toyoura sand; (b) No.7 silica sand and (c) No.8 silica sand.
Figure 11. Initial tangent Poisson’s ratio υi versus methane hydrate saturation Sh for specimens formed by (a) Toyoura sand; (b) No.7 silica sand and (c) No.8 silica sand.
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Figure 12. Coefficient f versus initial tangent Poisson’s ratio υi. The solid line is calculated using Equation (28).
Figure 12. Coefficient f versus initial tangent Poisson’s ratio υi. The solid line is calculated using Equation (28).
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3.7. Unloading Modulus

The unloading modulus Eur is often 2 to 3 times larger than the initial tangent modulus Ei for many geomaterials [21]. Miyazaki et al. [16] conducted unloading-reloading test on hydrate-sand specimen formed by Toyoura sand under 1 MPa effective confining pressure to obtain Eur. Figure 13 shows the Eur and Ei under 1 MPa effective confining pressure plotted against methane hydrate saturation Sh. The solid line denotes the calculated ei using Equations (13) and (15–17) for Toyoura sand specimen and the broken lines denote k-times the calculated ei. It is found that the Eur of hydrate-sand specimen is approximately 3 to 3.5 times Ei under 1 MPa effective confining pressure.
Figure 13. Unloading modulus Eur and initial tangent modulus Ei versus methane hydrate saturation Sh for specimens formed by Toyoura sand under 1 MPa effective confining pressure. The solid line denotes ei calculated using Equations (13) and (15–17). The broken lines are k-times the calculated ei.
Figure 13. Unloading modulus Eur and initial tangent modulus Ei versus methane hydrate saturation Sh for specimens formed by Toyoura sand under 1 MPa effective confining pressure. The solid line denotes ei calculated using Equations (13) and (15–17). The broken lines are k-times the calculated ei.
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4. Conclusions

In this study, a nonlinear elastic model of the mechanical behavior of artificial methane-hydrate-bearing sediment samples was developed. The presented model is based on the Duncan-Chang model and considered the dependences of the mechanical properties on methane hydrate saturation and effective confining pressure on the basis of the experimental results reported in an earlier work [19]. Some parameters were decided depending on the type of sand forming a specimen. In addition, the plot of lateral strain versus axial strain was also formulated as a function of effective confining pressure. Volumetric strain has not been discussed in this paper. It may be possible to associate the volumetric strain calculated using the presented model with the volumetric strain measured in the experiments.
The presented model is very easy to introduce into finite element codes. The applicable range of the presented model will be validated by numerical simulations for the laboratory-scale methane hydrate dissociation experiments.

Acknowledgments

This work was financially supported by the Research Consortium for Methane Hydrate Resources in Japan (MH21 Research Consortium) in the Japan’s Methane Hydrate R&D Program by the Ministry of Economy, Trade and Industry (METI). We appreciate insightful discussions with Akira Masui. We thank Takao Ohno and Shigenori Nagase for assistance with the experiments.

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Miyazaki, K.; Tenma, N.; Aoki, K.; Yamaguchi, T. A Nonlinear Elastic Model for Triaxial Compressive Properties of Artificial Methane-Hydrate-Bearing Sediment Samples. Energies 2012, 5, 4057-4075. https://doi.org/10.3390/en5104057

AMA Style

Miyazaki K, Tenma N, Aoki K, Yamaguchi T. A Nonlinear Elastic Model for Triaxial Compressive Properties of Artificial Methane-Hydrate-Bearing Sediment Samples. Energies. 2012; 5(10):4057-4075. https://doi.org/10.3390/en5104057

Chicago/Turabian Style

Miyazaki, Kuniyuki, Norio Tenma, Kazuo Aoki, and Tsutomu Yamaguchi. 2012. "A Nonlinear Elastic Model for Triaxial Compressive Properties of Artificial Methane-Hydrate-Bearing Sediment Samples" Energies 5, no. 10: 4057-4075. https://doi.org/10.3390/en5104057

APA Style

Miyazaki, K., Tenma, N., Aoki, K., & Yamaguchi, T. (2012). A Nonlinear Elastic Model for Triaxial Compressive Properties of Artificial Methane-Hydrate-Bearing Sediment Samples. Energies, 5(10), 4057-4075. https://doi.org/10.3390/en5104057

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