6th International Conference on Gas Hydrates

VARIABLE-COMPLIANCE-TYPE CONSTITUTIVE MODEL FOR METHANE HYDRATE BEARING SEDIMENT Miyazaki, Kuniyuki; Masui, Akira; Haneda, Hironori; Ogata, Yuji; Aoki, Kazuo; Yamaguchi, Tsutomu 2008-07-31

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Proceedings of the 6th International Conference on Gas Hydrates (ICGH 2008), Vancouver, British Columbia, CANADA, July 6-10, 2008.  VARIABLE-COMPLIANCE-TYPE CONSTITUTIVE MODEL FOR METHANE HYDRATE BEARING SEDIMENT Kuniyuki Miyazaki, Akira Masui, Hironori Haneda, Yuji Ogata, Kazuo Aoki and Tsutomu Yamaguchi National Institute of Advanced Industrial Science and Technology Onogawa, Tsukuba 305-8569 JAPAN ABSTRACT In order to evaluate a methane gas productivity of methane hydrate reservoirs, it is necessary to develop a numeric simulator predicting gas production behavior. For precise assessment of longterm gas productivity, it is important to develop a mathematical model which describes mechanical behaviors of methane hydrate reservoirs in consideration of their time-dependent properties and to introduce it into the numeric simulator. In this study, based on previous experimental results of triaxial compression tests of Toyoura sand containing synthetic methane hydrate, stress-strain relationships were formulated by variable-compliance-type constitutive model. The suggested model takes into account the time-dependent property obtained from laboratory investigation that time dependency of methane hydrate bearing sediment is influenced by methane hydrate saturation and effective confining pressure. Validity of the suggested model should be verified by other laboratory experiments on time-dependent behaviors of methane hydrate bearing sediment. Keywords: methane hydrate, triaxial compression test, stress-strain curve, strength, elastic modulus, strain rate, time dependency, constitutive equation NOMENCLATURE a1, a3, m1, m3 Parameters in compliance-variabletype constitutive model C Strain rate in constant strain rate test C1 Lower strain rate in alternating strain rate test C2 Higher strain rate in alternating strain rate test Ei Initial elastic modulus [MPa] E50 Secant elastic modulus at 50 % failure [MPa] n, n1, n3 Parameters determining time-dependency [-] Rc2 Increasing rate of strength when strain rate is doubled [-] Sh Methane hydrate saturation [%] t Time [sec] tc Time from the start of loading until strain reaches (σc / Ei) [sec] tc0 Given time [sec] ε Axial strain   λ Compliance [/MPa] λ0 Initial compliance [/MPa] λ* Normalized compliance [-] σ Differential stress [MPa] σc Peak strength in constant strain rate test [MPa] σc0 Peak strength at a certain strain rate [MPa] σc1 Lower peak strength in alternating strain rate test [MPa] σc2 Higher peak strength in alternating strain rate test [MPa] σ3’ Effective confining pressure [MPa] σ* Stress severity [-] INTRODUCTION Methane hydrate is anticipated to be a promising energy resource of natural gas, since a large amount of reservoir exists in marine sediments or in permafrost regions worldwide [1-3]. For the  Corresponding author: Phone: +81 29 861 8753 Fax +81 29 861 8765 E-mail: miyazaki-kuniyuki@aist.go.jp  purpose of efficient extraction of natural gas from the methane hydrate reservoirs, some methods to dissociate hydrate in-situ have been proposed; depressurization, thermal stimulation and inhibitor injection. In order to evaluate methane gas productivity from the reservoirs, it is essential to develop a numeric simulator including formation and dissociation behavior of methane hydrate, thermal properties of the reservoirs, permeability and mechanical behaviors of the reservoirs and so on. For reliable simulation of long-term behavior, it is important to predict mechanical behaviors of the reservoirs such as consolidation and deformation in consideration of their time-dependent properties. Therefore, it is necessary to develop a mathematical model which describes constitutive relationship (stress-strain relationship) of methane hydrate bearing sediment including their timedependent behaviors and to introduce it into the numeric simulator predicting gas production behavior. In this study, based on previous experimental results of triaxial compression tests of Toyoura sand containing synthetic methane hydrate [4,5], stress-strain relationships were formulated by variable-compliance-type constitutive model which has been applied to various time-dependent behaviors of rock [6,7]. EXPERIMENTAL METHOD Experimental method published previously can be summarized as follows [4,5,8].  such as hydrate formation, water substitution, axial compression and hydrate dissociation. The apparatus is a digital servo-controlled testing machine with a capacity of 200 kN for axial load, 20 MPa for confining pressure and 20 MPa for pore pressure. The temperature in the pressure vessel can be controlled at the range of 243 K to 293 K with an accuracy of 0.5 K by circulating confining refrigerant liquid from a cooling tank. Experimental data, such as axial load measured by a strain gauge-type load cell and axial displacement measured by a linear voltage differential transformer, were recorded by the data acquisition system at every second during the experiments. Hydrate formation and water substitution Two kinds of specimen were tested: one is hereafter called “saturated-sand specimen” and the other “hydrate-sand specimen.” The former contained no methane hydrate and the latter contained synthetic methane hydrate in a variety of degrees. After a host specimen was set with top cap, rubber sleeve and pedestal as shown in Figure 1, methane gas was percolated into a host specimen to replace air existing in pore space at temperature of 278 K. Then pore pressure was increased up to the induction pressure of 8.0 MPa at the rate of 0.5 MPa to 1.0 MPa per minute,  1  2  3  10 9  4  Preparation of host specimen A host specimen, in which synthetic methane hydrate was formed afterward, was prepared by compacting water-saturated Toyoura sand densely in a mold on a vibration table. The initial water content, which had a great influence on methane hydrate saturation of the specimen, was adjusted by draining excess water with a syringe pump. Then unsaturated sand was frozen in a freezing chamber so as to make it easy to handle the host specimen. The size of frozen host specimen was 50 mm in diameter and 100 mm in length and its porosity ranged from 37 % to 39 %. Experimental apparatus Experimental apparatus illustrated in Figure 1 was used in each process of triaxial compression test  5 6 7 8 11  3  12  1. Load cell 2. Displacement transducer 3. Refrigerant pipe 4. Pressure vessel 5. Top cap 6. Rubber sleeve 7. Specimen  13  8. Pedestal 9. Gas-water separator 10. Gas flow meter 11. Pressure gauge 12. Water cylinder 13. Gas cylinder  Figure 1 Experimental apparatus  12 10 Differential stress [MPa]  while confining pressure was increased at the same rate. Successively, pore pressure, confining pressure and temperature were kept constant during 24 hours of induction period. After that, water substitution was conducted by injecting water into the specimen to replace methane gas remaining in pore space in the specimen, pedestal, top cap, pipes and so on. The volume of injected water was almost twice as much as that of specimen.  σ 3' = 3 MPa 8 2 MPa  6  1 MPa  4  0.5 MPa  2 0 0  5  EXPERIMENTAL RESULTS Constant strain rate test Stress-strain curves shown in Figure 2 were obtained from constant strain rate tests. As shown in Figure 2 (a), in the case of saturated-sand specimen of methane hydrate saturation 0 %, stress increased and slope of the curve decreased until stress reached the peak strength at more than 5 % of strain. Both of the stress and strain at the  15  Axial strain [%] 12 10  σ 3' = 3 MPa  8  2 MPa  6 1 MPa 4 0.5 MPa 2 0 0  5  10  15  Axial strain [%]  (b) Sh = 25 % 12 σ 3' = 3 MPa  Differential stress [MPa]  10  After the triaxial compression test was carried out, methane hydrate formed in the specimen was dissociated by depressurizing pore space and the amount of released methane gas was measured so that the initial methane hydrate volume in the specimen would be calculated.  10  (a) Sh = 0 %  Differential stress [MPa]  Axial compression and hydrate dissociation Axial compression was carried out with a servocontrolled strain rate. Two control modes were conducted. In one mode, strain rate was kept constant 0.1 % per minute. In the other mode, strain rate was changed alternately between two speeds C1 and C2 at a constant strain interval 0.5 %. Hereafter, axial compression test conducted in the former and latter mode are called “constant strain rate test” and “alternating strain rate test” respectively. Since relatively sufficient data had been already obtained at constant strain rate of 0.1 % per minute, it was adopted as the higher strain rate C1 in alternating strain rate test. With reference to previous study [9], the lower strain rate C2 was decided to be 0.01 % per minute for saturated-sand specimen or 0.05 % per minute for hydrate-sand specimen. Axial compression was conducted at a temperature around 278 K in drained condition maintaining pore pressure of 8 MPa and confining pressure of 8.5 MPa, 9 MPa, 10 MPa or 11 MPa, and thus effective confining pressure was maintained at 0.5 MPa, 1 MPa, 2 MPa or 3 MPa during the axial compression process.  2 MPa  8 6  1 MPa  4 0.5 MPa 2 0 0  5  10  15  Axial strain [%]  (c) Sh = 35 % Figure 2 Stress-strain curves in constant strain rate tests  peak strength increased with effective confining pressure. In the region after the peak strength, the stress gradually decreased with strain. Figure 2 (b) and (c) shows stress-strain curves of hydrate-sand specimens of methane hydrate saturation 25 % and 35 % respectively. As shown in these figures, the stress-strain curves of hydrate-sand specimen were generally similar to those of saturated-sand specimen in terms of shape, though peak strength, elastic modulus and strain softening tendency varied with methane hydrate saturation. Peak strength was plotted against methane hydrate saturation in Figure 3. From the figure, peak strength increased with methane hydrate saturation. For example, in case of effective confining pressure 1 MPa, peak strength increased approximately two times with increase in methane hydrate saturation from 0 % to 55 %. Peak strength σc can be approximately expressed as the following function of methane hydrate saturation Sh and effective confining pressure σ3’: σc = 3.67 σ3’ 0.754 + 0.00249 Sh 1.86. (1) Elastic modulus, or secant elastic modulus at 50 % failure, was plotted against methane hydrate saturation in Figure 4. From the figure, elastic modulus increased with methane hydrate saturation. Elastic modulus E50 can be approximately expressed as the following function of methane hydrate saturation Sh and effective confining pressure σ3’: E50 = 246 σ3’ 0.482 + 10.8 Sh. (2)  Masui et al. noted that the increases in peak strength and elastic modulus with methane hydrate saturation were due to cementation between sand particles by methane hydrate [4, 5]. Alternating strain rate test In order to predict long-term mechanical behavior, it is essential to investigate time-dependent property. Loading rate dependency is one of important time-dependent behaviors. Loading rate dependency of hydrate-sand specimen was found to be significantly apparent, while that of saturated-sand specimen was negligible small [8, 9]. This result indicates that time-dependency of methane hydrate bearing sediment is strongly influenced by the methane hydrate saturation, which is likely because methane hydrate in pore space of sediment relates closely to deformation mechanism of time-dependent behaviors. In this study, loading rate dependency of saturated-sand specimen and hydrate-sand specimen of a variety of methane hydrate saturations was experimentally examined under conditions of effective confining pressure 0.5 MPa, 1 MPa, 2 MPa and 3MPa. As shown schematically in Figure 5, stress-strain curves obtained from alternating strain rate tests showed an increase/decrease of stress at an increase/decrease of strain rate. Curve 1 in Figure 5 is the spline curve connecting the points marked with ○ when strain rate was switched from C1 to C2 and can be considered as the stress-strain curve at constant strain rate C1. Likewise, Curve 2 is the  16  1200 σ 3' Exp. data 3 MPa ▲ 2 MPa □ 1 MPa ◆ 0.5 MPa ○  σ 3' Exp. data 3 MPa ▲ 2 MPa □ 1 MPa ◆ 0.5 MPa ○  Eq. (2)  Eq. (1)  12  1000 Elastic modulus [MPa]  Peak strength [MPa]  14  10 8 6 4  800 600 400 200  2 0  0 0  20  40  60  Methane hydrate saturation [%]  Figure 3 Peak strength in constant strain rate test  0  20  40  60  Methane hydrate saturation [%]  Figure 4 Elastic modulus in constant strain rate test  spline curve connecting the points marked with ● when strain rate was switched from C2 to C1 and can be considered as the stress-strain curve at constant strain rate C2. In this way, two stressstrain curves and thus two peak strengths σc1 and σc2 corresponding to different strain rates C1 and C2 were obtained from a single specimen. As shown in Figure 6, stress-strain curves obtained from alternating strain rate tests under effective confining pressure 1 MPa were generally similar to those from constant strain rate tests, though they undulated with switching strain rate. The amplitude of undulating curves seems to depend on methane hydrate saturation.  expressed by Equation (1) displayed in Figure 7. From the figure, the difference between σc1 and σc2 widened as methane hydrate saturation increased. For hydrate-sand specimen, the increasing rate of strength Rc2 can be calculated by the following expression: Rc2 = (σc2 / σc1) – 1. (3) Because the ratio of strain rate C2 / C1 was set to be 2 for hydrate-sand specimen, Equation (3) is regarded as giving the increasing rate of strength when strain rate is doubled. Relations between increasing rate of strength Rc2 and methane hydrate saturation under various effective confining 8 S h = 48 % Differential stress [MPa]  Strengths obtained from alternating strain rate tests of saturated-sand and hydrate-sand specimens under various effective confining pressures were shown in Figure 7. In this figure, two coupled marks represent a pair of strengths obtained from one specimen as schematically shown in the inset. Relations between strength σc1 and methane hydrate saturation Sh under each effective confining pressures σ3’ were approximately  6 20 % 4  0%  2  Experimental result Curve 1 Curve 2  Curve 2 σ c2  0 0  Curve 1  Stress-strain curve Point of switching strain rate from C 1 to C 2 Point of switching strain rate from C 2 to C 1  Axial strain  Strain rate  5  10  15  Axial strain [%]  Figure 6 Stress-strain curves in alternating strain rate tests 16 σ 3' Exp. data 3 MPa ▲ 2 MPa □ 1 MPa ◆ 0.5 MPa ○  Eq. (1)  14 Peak strength [MPa]  Differential stress  σ c1  12 10 8 6 4  C2  σ c1  2  C1  σ c2  0  Strain interval (= 0.5 %) Axial strain  Figure 5 Schematic stress-strain curve in alternating strain rate test  0  20  40  60  Methane hydrate saturation [%]  Figure 7 Peak strength in alternating strain rate test  0.12  CONSTITUTIVE MODEL Based on the experimental results above, applicability of variable-compliance-type constitutive model, which focuses on timedependent behaviors such as strain rate dependency of strength, to methane hydrate bearing sediment was examined. Variable-compliance-type constitutive model In this study, variable-compliance-type constitutive model was adopted to simulate stressstrain relationships of saturated-sand and hydratesand specimen in constant strain rate tests. The model is applicable to various time-dependent behaviors of rock including strain rate dependency of peak strength. It was previously reported that cohesion or adhesive characteristics of sand was enhanced in the presence of methane hydrate by its cementing effect between sand particles [4,5]. Therefore, assuming that methane hydrate bearing sediment takes on somewhat similar properties with rock from a mechanical point of view, it is expected that time-dependent behaviors of methane hydrate reservoirs can be expressed by the model. This is a key reason why the model was decided to be examined in this study.  Increasing rate of strength [ - ]  pressures were shown in Figure 8. They were linearly related and the degree of effect of methane hydrate saturation on increasing rate of strength depended on effective confining pressure. Considering that the increasing rate of strength of saturated-sand specimen varied little with confining pressure as described later, increasing rate of strength Rc2 can be approximately expressed as the following function of methane hydrate saturation Sh and effective confining pressure σ3’: Rc2 = 0.00189 σ3’ -0.333 Sh + 0.0072. (4) The constant term in right side of Equation (4) was decided on experimental results of the increasing rate of strength of saturated-sand specimen as described later.  dλ * dt λ* =  = a 1 ( λ * - 1) λ λ0  ,λ =  ε σ  - m1  σ*  , λ0 =  n1  1 Ei  +a3 λ * , σ* =  m3  n  σ* 3,  σ σ c0  (5) ,  where t is time, λ is compliance, λ* is normalized compliance, λ0 is initial compliance or inverse of  0.08 0.06 0.04  σ 3' Exp. data 3 MPa ▲ 2 MPa □ 1 MPa ◆ 0.5 MPa ○  Eq. (4)  0.02 0 0  20  40  60  Methane hydrate saturation [%]  Figure 8 Increasing rate of strength initial elastic modulus, ε is strain, σ is principal stress difference, Ei is initial elastic modulus, σ* is stress severity or inverse of local safety factor, σc0 is peak strength not at strain rate 0.1 % per minute but at a certain strain rate given later and a1, m1, n1, a3, m3 and n3 are model parameters. Among parameters in Equation (5), n1 and n3 indicate time-dependency or viscoelasticity. However, the difference between n1 and n3 has not been well understood actually. In this study, it was assumed that there is no difference between n1 and n3 [6,10]: n = n1 = n3. (6) Given Equation (6), the differential equation in Equation (5) yields: dλ * dt  = { a 1 ( λ * - 1)  - m1  + a3 λ *  m3  n  }•σ * .  (7)  From the solution of Equation (7) under constant strain rate C, it turns out that peak strength σc is proportional to the 1 / (n+1) power of C [7]. Thus, if peak strengths σc1 and σc2 corresponding to different strain rates C1 and C2 are known, the value of parameter n can be obtained from the following equation: n =  Variable-compliance-type constitutive model is expressed as follows:  0.1  log( C 2 / C 1 ) log( σ c2 / σ c1 )  - 1.  (8)  For example, the values of (σc2 / σc1) for saturatedsand specimens were independent of effective confining pressure σ3’ and the average value was 1.0241 from results of alternating strain rate tests in which the ratio of strain rate (C2 / C1) was set to be 10. When 1.0241 and 10 are assigned to (σc2 / σc1) and (C2 / C1) respectively in Equation (8), 95.6 is obtained as the value of n for saturated-sand  specimen. Therefore 1.0072 can be obtained as the value of (σc2 / σc1) by solving Equation (8) in which 95.6 and 2 are assigned to n and (C2 / C1) respectively. The constant term 0.0072 in right side of Equation (4), or the increasing rate of strength Rc2 for saturated-sand specimen when strain rate was doubled, was obtained by assigning this value 1.0072 to the (σc2 / σc1) in Equation (3).  10 a 1 = 0.001  Differential stress [MPa]  8  Decision of parameters Given Equation (6), seven parameters, Ei, σc0, a1, m1, a3, m3 and n, should be determined according to experimental results. In this study, these parameters were determined in the way hereinafter described.  a 1 = 0.01 a 1 = 0.1  6  4 Experimental result (S h = 40 %, σ 3' = 1 MPa)  2  0 0  5  10  15  Axial strain [%]  Secant elastic modulus at 50 % failure E50 was adopted as initial elastic modulus Ei in Equation (5): Ei = E50 = 246 σ3’ 0.482 + 10.8 Sh. (9) The parameter n can be calculated by Equation (8). Firstly Equation (3) yields the following expression: (σc2 / σc1) = Rc2 + 1. (10) Then the following equation can be derived by assigning 2 and Equation (10) to (C2 / C1) and (σc2 / σc1) respectively: n=  1 log  2  ( R c2 + 1)  (11)  - 1.  Thus the following equation can be derived from Equation (4) and (11): 1  n= log  2  ( 0.00189 σ 3 '  - 0.333  S h + 1.0072 )  - 1 . (12)  Figure 9 Variation of stress-strain curve with the value of a1 (13) with Equation (1), (9), (12) and assignment of the following expressions: tc0 = 120 seconds, (14) C = 0.1 % per minute. (15) The other parameters, a1, m1, a3 and m3 were determined as bellow so that the numerical solution fitted the experimental results of constant strain rate tests shown in Figure 2. The term which includes a3 and m3 in Equation (7) is dominant mainly in the region after the peak strength. So the determination of a1 and m1 started with an assumption that a3 equaled 0, in reference to the stress-strain curves in the region before the peak strength: dλ *  Because peak strength depends on strain rate, in order to determine the value of σc0, it is necessary to decide the strain rate C0 corresponding to σc0. In this study, C0 is determined to be the strain rate which meets the requirement that the time tc from the start of loading until strain reaches (σc / Ei) equals a given constant tc0. In this instance, the following equation can be derived from the solution of Equation (7) under constant strain rate C: σ c0 = (  tc t c0  1  )n σc = (  σ c / Ei / C t c0  1  )n σc.  (13)  The time tc0 can be determined arbitrarily. In this study, tc0 was determined to be 120 seconds. The peak strength σc0 can be calculated from Equation  dt  = a 1 ( λ * - 1)  - m1  n  σ* .  (16)  The parameter m1 indicates mainly the shape of stress-strain curve in the region before the peak strength. In accordance with all of the experimental results in this study, it was decided that the value of m1 can be expressed as follows: m1 =  n +1 4 . 90  - 1.  (17)  For example, experimental and calculated stressstrain curves of hydrate-sand specimen of methane hydrate saturation 40 % under effective confining pressure 1 MPa were shown in Figure 9. Calculated curves in Figure 9 were obtained with a variety of a1 in Equation (16). The value of a1 was  In case that a3 equals 0, the calculated result does not sufficiently express the experimental residual strength or stress-strain relationship in the region after peak strength. To return to Equation (7), a3 and m3 were determined in reference to stressstrain curves in the region after the peak strength.  12 10 Differential stress [MPa]  determined so that the calculated curve passed slightly above the experimental curve: a1 = 0.01. (18)  σ 3' = 3 MPa 8 2 MPa 6 1 MPa  4  0.5 MPa 2  The slope of stress-strain curve after the peak strength depends on m3. In this study, the value of m3 was determined as follows: m3 = 2.5. (19)  0 0  It can be said that variable-compliance-type constitutive model is appreciably simple, because compliance λ, or inverse of elastic modulus, is the only variable which varies with time t following Equation (5). Since one has only to incorporate a subroutine for variation of elastic modulus into a simulation code for elastic analysis, the model is easy to introduce into a numerical simulation of  15  Axial strain [%]  12 σ 3' = 3 MPa  Differential stress [MPa]  10 8  2 MPa  6 1 MPa 4 0.5 MPa 2 0 0  5  10  15  Axial strain [%]  (b) Sh = 25 % 12 σ 3' = 3 MPa 10 Differential stress [MPa]  As above, all of the parameters in Equation (5) corresponding to various methane hydrate saturations and effective confining pressures were determined with Equation (1), (6), (9), (12-15) and (17-20). The stress-strain relationships under constant strain rate 0.1 % per minute can be numerically calculated by these equations. Calculated curves (with marks) and experimental curves (without marks) were shown in Figure 10. It was found that experimental stress-strain curves under various methane hydrate saturations and effective confining pressures were approximately expressed by the model under corresponding conditions. This result indicates that variablecompliance-type constitutive model can be considered to be applicable to mechanical behavior of methane hydrate bearing sediment under triaxial stress state to a considerable extent.  10  (a) Sh = 0 %  The parameter a3 decides the peak strength. In this study, the value of a3 was determined as follows: a3 = 0.0001. (20) Equation (18), (19) and (20), which were derived from in accordance with all of the experimental results in this study, suggested that a1, a3 and m3 were almost independent of methane hydrate saturation or effective confining pressure.  5  2 MPa  8 6  1 MPa  4  0.5 MPa  2 0 0  5  10  15  Axial strain [%]  (c) Sh = 35 % Figure 10 Experimental and calculated stressstrain curves  mechanical behavior. Therefore the model is considered sufficiently promising because it can assess long-term behaviors of methane hydrate bearing sediments. The validity of the model will be appreciated by creep tests or other laboratory experiments concerning time-dependency. CONCLUSIONS In this study, applicability of variable-compliancetype constitutive model based on the results of constant and alternating strain rate tests to mechanical behavior of methane hydrate bearing sediment was examined. As a result, it was found that the model was sufficiently applicable to stress-strain relationships of saturated-sand and hydrate-sand specimens under triaxial stress state. One of important issues toward putting the model to practical use is to verify its validity by other mechanical experiments on time-dependent behaviors of methane hydrate bearing sediment. ACKNOWLEDGEMENTS The research has been conducted in the project of MH21 research consortium. Thanks are due to our sponsor and/or partners. Our special thanks and warm appreciation should be expressed to Mr. Takao Ohno and Mr. Shigenori Nagase, technical staffs of AIST. REFERENCES [1] Kvenvolden KA. A Major Reservoir of Carbon in the Shallow Geosphere? Chem Geol 1988; 71(1-3): 41-51. [2] Kvenvolden KA, Ginsburg GD, Soloviev VA. Worldwide Distribution of Subaquatic Gas Hydrates. Geo-Mar Lett 1993; 13(1): 32-40. [3] Okuda Y. Petrotech 1993; 16: 300-306. (in Japanese) [4] Masui A, Haneda H, Ogata Y, Aoki K, The effect of The Effect of saturation degree of methane hydrate on the shear strength of synthetic methane hydrate sediments: ICGH, Proc 5th Int Conf on Gas Hydrates, Trondheim, 2005. [5] Masui A, Haneda H, Ogata Y, Aoki K. Effects of Methane Hydrate Formation on Shear Strength of Synthetic Methane Hydrate Sediments: ISOPE, Proc 15th Int Offshore and Polar Eng Conf, Seoul, 2005. [6] Okubo S, Fukui K, Hashiba K. Extension of Constitutive Equation of Variable Compliance Type and its Validation based on Creep Data. J MMIJ 2002; 118(12): 737-744. (in Japanese)  [7] Okubo S, Fukui K, Hashiba K. Extension of Constitutive Equation of Variable Compliance Type and its Validation based on Uniaxial Compression and Tension Tests of Sanjome Andesite. J MMIJ 2003; 119(9): 541-546. (in Japanese) [8] Miyazaki K, Masui A, Haneda H, Ogata Y, Aoki K, Yamaguchi T, Variable-Compliance-Type Constitutive Model for Toyoura Sand Containing Methane Hydrate: ISOPE, Proc 7th ISOPE Ocean Mining & Gas Hydrates Symposium, Lisbon, 2007. [9] Miyazaki K, Masui A, Sakamoto Y, Haneda H, Ogata Y, Aoki K, Yamaguchi T, Okubo S. Strain Rate Dependency of Sediment Containing Synthetic Methane Hydrate in Triaxial Compression Test. J MMIJ 2007; 123(11): 537544. (in Japanese) [10] Okubo S, Fukui K. Long-Term Creep Test and Constitutive Equation of Tage Tuff. J MMIJ 2002; 118(1): 36-42. (in Japanese)  

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